Geometry Problems CollectionMarch, 2005

me crazywong

1.Let ABCDEF be a hexagon with AB = BC = CD, DE = EF = FA and 6 BCD = 6 EFA = 60◦. Let Gand H be two points inside the hexagon such that 6 AGB = 6 DHE = 120◦. Show that

AG + GB + GH + DH + HE ≥ CF

2.Let ABC be a triangle with 6 B = 60◦, AD the altitude, and H the orthocenter. Show that the circumcenterO lies on the angle bisector of 6 DHC.

3.Two congruent circles intersect at points A and B. Two more circles of the same radius are drawn: onethrough A, the other through B. Prove that the four points of the paired intersection of all four circles (otherthan A and B) are the vertices of a parallelogram.

4.Let O be the circumcenter and H the orthocenter of an acute 4ABC. Prove that the area of one of the4AOH, 4BOH and 4COH is equal to the sum of the areas of the other two.

5.On the plane, a line l intersect sides AB and AC of 4ABC at D and E respectively, such that AD + AE :AB + BC + CA = [ADE] : [ABC]. Show that l passes through the incenter of 4ABC.

6.Suppose ABCD is a square piece of cardboard with side length a. On a plane are two parallel lines `1 and`2, which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect`1 at E and F respectively. Also, sides CB and CD intersect `2 at G and H respectively. Let the perimetersof 4AEF and 4CGH be m1 and m2 respectively. Prove that no matter how the square was placed, m1 + m2

remains constant.

7.(a)In 4ABC, D, E, F are points inside 4ABC such that AD bisects 6 A, BE bisects 6 B and CF bisects6 C. Prove that

AB ·BC · CA > AD ·BE · CF

(b)Instead of angle bisectors, if AD, BE, CF are the medians, and a >√

2|b− c|, b >√

2|c− a|, c >√

2|a− b|.Prove that the inequality in (a) is still true.

8.In 4ABC, P and Q are on AB, R is on CA. If P , Q, R trisect the perimeter of 4ABC, prove that

9[PQR] > 2[ABC]

9.Given that P is a point lying outside circle O, PS and PT are tangents to circle O. Through P draw a linemeeting circle O at points A and B, let PAB meet ST at C. Show that

1PC

=12(

1PA

+1

PB)

By mecrazywong

1

10.In a quadrilateral ABCD, AB = BC = AD, 6 ABC = 60◦. A point P is inside ABCD, prove that

PA + PD + PC ≥ BD

11.Let ABCD be a convex quadrilateral, P , Q and R be the feet of the prependiculars from D to the linesBC, CA and AB respectively. Show that PQ = QR if and only if the bisectors of 6 ABC and 6 ADC meet onAC.

12.In a convex quadrilateral ABCD the diagonal BD bisects neither the 6 ABC nor the 6 CDA. A point P liesinside ABCD and satisfies 6 PBC = 6 DBA and 6 PDC = 6 BDA. Prove that ABCD is a cyclic quadrilateralif and only if AP = CP .

13.Let ABC be an acute-angled triangle with AB 6= AC. The circle with diameter BC intersects the sides ABand AC at M and N respectively. Denote by O the midpoint of the side BC. The bisectors of the 6 BAC and6 MON intersect at R. Prove that the circumcircles of 4BMR and 4CNR have a common point lying onthe side BC.

14.In 4ABC, AB = BC. A line through B cuts AC at D so that radius of the incircle of 4ABD equals that

of the excircle of 4CBD opposite B. Prove that this radius ish

4, where h is the altitude from C to AB.

15.Two circles of radii a and b touch each other externally. Let c be a radius of a circle that touches these twocircles as well as common tangent to the two circles. Prove that

1√c

=1√a

+1√b

16.(Euler Line) Prove that in any triangle the circumcenter O, orthocenter H and centroid G are collinear andOG : GH = 1 : 2.

17.(Fermat’s Point) Given that a point P inside 4ABC with the greatest angle less than 120◦. Prove thatPA + PB + PC is minimum if and only if 6 APB = 6 BPC = 6 CPA = 120◦.

18.(Ptolemy’s Theorem) Prove that for any convex quadrilateral ABCD, AB ·CD + BC ·AD ≥ AC ·BD andequality holds if and only if ABCD is concyclic.

19.(Morley’s Theorem) Prove that the points of intersection of the nine angle trisectors of a triangle forms anequilateral trianlge.

20.Given a 4ABC with 6 B = 30◦ and shortest side AC. Points D and E are on BA and BC respectivelysuch that AD = CE = AC. Prove that OI = DE, OI ⊥ DE and O is the orthocenter of 4DEI, where Oand I are the circumcenter and incenter of 4ABC respectively.

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21.(Euler’s Theorem) Let d be the distance between the circumcenter and incenter of a triangle. Show that

d2 = R(R− 2r)

where R and r are the circumradius and inradius respectively.

22.In a convex hexagon ABCDEF , each of AD, BE and CF bisects its area. Prove that AD, BE and CFare concurrent.

23.I is the incenter of 4ABC. N and M are the midpoints of sides AB and CA respectively. The lines BIand CI meet MN at K and L respectively. Prove that AI + BI + CI > BC + KL.

24.Let D and E be the points on side BC of 4ABC such that 6 BAD = 6 CAE. Let the incircles of 4ABDand 4ACE touch BC at M and N respectively. Show that

1MB

+1

MD=

1NC

+1

NE

25.Let P be a point inside triangle ABC such that

6 APB − 6 ACB = 6 APC − 6 ABC.

Let D,E be the incenters of triangles APB, APC, respectively. Show that AP,BD, CE meet at a point.

26.The angle at A is the smallest angle of 4ABC. The points B and C divide the circumcircle of the triangleinto two arcs. Let U be an interior point of the arc between B and C which does not contain A. Theperpendicular bisectors of AB and AC meet the line AU at V and W , respectively. The lines BV and CWmeet at T . Show that

AU = TB + TC

27.ABC is a triangle; P , Q, R are points outside 4ABC such that ARB, BPC and CQA are similar isoscelestriangles with the sides of 4ABC as bases. Show that AP , BQ and CR are concurrent.

28.In a 4ABC, 6 C = 26 B. P is a point inside 4ABC such that AP = PC and PB = PC. Prove that6 PAC = 26 PAB.

29.Prove that in a 4ABC 6 A = 26 B if and only if BC2 = AC(AC + AB).

30.A quadrilateral ABCD is concyclic, AC and BD meet at P . Let O be the circumcenter of 4APB and Hthe orthocenter of 4CPD. Prove that O, P and H are collinear.

31.ABCD is a cyclic quadrilateral, AC ⊥ BD. Let AC and BD meet at E. Show that

EA2 + EB2 + EC2 + ED2 = 4R2

where R is the circumradius.By mecrazywong

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32.Let ABC be an acute-angled triangle. AP and AQ are tangents to the circle with BC as the diameter.Prove that the orthocenter of 4ABC lies on PQ.

33.(Bretschneider’s Formula and Brahmagupta’s Formula) Denote S by the area of a quadrilateral of sidesa, b, c, d. Show that

S =

√(s− a)(s− b)(s− c)(s− d)− abcd cos2

B + D

2=

14

√4p2q2 − (b2 + d2 − a2 − c2)2

where p, q are the diagonals and s is the semiperimeter.

34.(Strong Ptolemy’s Theorem) Let ABCD be a quadrilateral. Show that

AC2 ·BD2 = AB2 · CD2 + BC2 ·DA2 − 2AB ·BC · CD ·DA cos(B + D)

35.Let ABC be a triangle, X, Y , Z the centers of the external squares constructed with sides BC, CA, ABrespectively. Show that Y Z ⊥ AX and Y Z = AX.

36.Three fixed circles pass through points A and B. X is a variable point on the first circle different from A

and B. The line AX meets the other two circles at Y and Z (with Y between X and Z). Show that XYY Z

remains constant.

37.Let ABCD be a rectangle with M , N be the midpoints of AD and BC respectively. Extend CD to anypoint P . Let PM and AC meet at Q. Show that MN bisects 6 QNP .

38.Let D be an arbitrary point on side BC of 4ABC. Show that

sin 6 BAC

AD=

sin 6 BAD

AC+

sin 6 CAD

AB

39.(Stewart’s Theorem) Let D be an arbitrary point on side BC of 4ABC. Show that

AB2 · CD + AC2 ·BD −AD2 ·BC = BC ·BD ·DC

40.(Pascal’s Theorem) If A,B,C, D, E, F are six arbitrary points on a circle, show that the point of intersectionof the lines AB and DE, the point of intersection of the lines BC and EF , and the point of intersection of thelines CD and FA are collinear.

41.Let E,F,G be three fixed points on line l, A an arbitray point on circle O. Let AE, AG and FD meetcircle O at B, D and C respectively. Show that BC passes through a fixed point on l.

By mecrazywong

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42.Denote S by the area of a triangle ABC. Show that

S =12aha =

12ab sinC =

abc

4R= rs =

√s(s− a)(s− b)(s− c) = 2R2 sinA sinB sinC =

12R2(sin 2A+sin 2B+sin 2C)

43.Two circles, C1 and C2, centered at O1 and O2 respectively, meet at A and B. O1B is produced to meetC2 at E. O2B is produced to meet C1 at F . A straight line is constructed through B parallel to EF cuttingC1 and C2 at M and N respectively. Prove that

MN = AE + AF

44.Let ABC be a triangle, BE and CF the altitudes. Show that the perpendicular bisector of EF , the bisectorsof 6 ABE and 6 ACF are concurrent.

45.Let BCDE and ACFG be the two external squares constructed on the sides BC and AC of 4ABC. LetM be the midpoint of EG. Show that M lies on the perpendicular bisector of AB.

46.Let P and Q be the centers of the two external squares constructed on the sides BC and AC of 4ABC.Let M be the midpoint of AB. Show that PM = QM and 6 PMQ = 90◦.

47.Let O and H be the circumcenter and orthocenter of an acute-angled 4ABC, respectively. Let D and Ebe the points on AB and AC respectively, such that AD = AH and AE = AO, show that DE = AE.

48.Let AD be the diameter of the cicumcircle of an acute-angled 4ABC. Extend BC such that it meets thetangent passing through D at point P . Let PO meet sides AC and AB at points M and N respectively. Showthat OM = ON .

49.Let ABC be a triangle and D the foot of the altitude from A. Let E and F be on a line through D suchthat AE ⊥ BE, AF ⊥ CF , and E and F are different from D. Let M and N be the midpoints of the linesegments BC and EF , respectively. Prove that AN ⊥ NM .

50.Let ABEF and ACGH be the external squares constructed on the sides AB and AC of4ABC, respectively.Let AD be the altitude. Show that AD, BG and CE are concurrent.

51.Let points B′ and C ′ be the projections of points B and C on the exterior bisector of 6 A of 4ABC. LetAD be the angle bisector of 6 A. Show that AD, BC ′ and CB′ are concurrent.

52.Let ABC be a triangle with CA + AB = 2BC. Prove that the external bisector of 6 CAB is parallel to theline formed by the incenter and the circumcenter of 4ABC.

53.(Generalization of Problem 52) Let I and O be the incenter and circumcenter of 4ABC, respectively.Assume 4ABC is not equilateral. Show that 6 AIO ≤ 90◦ if and only if 2BC ≤ AB + CA.

By mecrazywong

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54.Let ABC be a triangle with no right angles, and let D be a point on side BC. Let E and F be the feet ofthe perpendiculars from D to AB and AC respectively. Let P be the point of intersection between BF andCE. Show that AP is the altitude of 4ABC if and only if AD is the interior angle bisector of 4ABC.

55.(Nine-point Circle) Show that in any triangle ABC, the feet of three altitudes, the midpoints of three sides,and the midpoints of the segments from the vertices to the orthocenter, all lie on the same circle. This circleis of radius half as the circumradius of triangle ABC.

56.Given a triangle ABC, prove that there exists a unique point P on the same plane such that

AP 2 + BC2 = BP 2 + CA2 = CP 2 + AB2 = 4R2

where R denotes the circumradius of 4ABC.

57.Prove that the nine-point center N of a triangle ABC lies on the Euler line, and it is the midpoint betweenthe circumcenter and the orthocenter of 4ABC.

58.Let ABCD be a cyclic quadrilateral, its opposite sides meet at points E and F . From E and F draw thetangents EG and FH to the circumcircle of quadrilateral ABCD. Prove that EF 2 = EG2 + FH2.

59.(Napoleon’s Theorem) Let ABF , BCD and ACE be the external equilateral triangles construct with sidesAB, BC and CA of 4ABC. Show that the centers of the three equaliteral triangles are the vertices of aequaliteral triangle. Show that it is also true for ABF , BCD and ACE being internal equilateral triangles,and the difference in positive of the areas of the equilateral triangles formed internally and externally equalsto the area of 4ABC.

60.(Carnot’s Theorem) Show that in any triangle the sum of the distances from the circumcenter to the threesides equals to the sum of circumradius and the inradius.

61.(Steiner’s Theorem) Let ra, rb, rc, R and r be the exradii, circumradius and inradius of 4ABC, respectively.Show that

ra + rb + rc = 4R + r

62.(Eyeball Theorem) Given two circles C1 and C2 centered at O1 and O2 respectively, such that O1 is outsideC2 and O2 is outside C1. Let the tangents from O1 to C2 meet C1 at points A1 and B1, the tangents from O2

to C1 meet C2 at points A2 and B2. Show that A1B1 = A2B2.

63.(Aubel’s Theorem) Let ABCD be a quadrilateral. Let P , Q, R, S be the centers of the external squaresconstructed on the sides AB, BC, CD, DA respectively. Show that PR = QS and PR ⊥ QS.

64.Let ABDE and BCGH be the external squares constructed on the sides AB and BC of 4ABC. Let Mbe the midpoint of AC = 2BM and AC ⊥ BM .

65.Let ABDE, BCGH, ACPQ be the external squares constructed on the sides AB, BC and AC of 4ABC.Let M be the midpoint of EG. Show that MD = MH, MD ⊥ MH and AMCO is a square.

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66.Let X, Y be the intersection points between the tangent line of the nine-point circle of 4ABC at themidpoint of BC with CA, AB respectively. Prove that B,C,X, Y are concyclic.

67.Let I be the incenter of 4ABC, IA, IB , IC the excenters of 4ABC. Show that the midpoints of IIA, IIB ,IIC , IAIB , IBIC , ICIA all lie on the circumcircle of 4ABC.

68.(Nagel Point) Let the three excircles of 4ABC touch sides AB, BC and CA at points F , D and Erespectively. Show that AD, BE and CF are concurrent.

69.Let P be a arbitrary point on the circumcircle of square ABCD. Let R be the circumradius. Show that

PA2 + PB2 + PC2 + PD2 = 8R2

70.Let AD, BE and CF be the altitudes of 4ABC. The projections of point D on AB, BE, CF , CA are M ,N , P , Q respectively. Prove that M , N , P , Q are collinear.

71.Let ABCD be a cyclic quadrilateral with 6 D = 90◦. The projections of point B on AC and AD are E andF respectively. Show that EF passes through the midpoint of BD.

72.Let D, E, F , G be the projections of point A on the internal and external bisectors of 6 ABC and 6 ACBof 4ABC. Prove that E, F , G, D are collinear.

73.Given three congruent circles centered at A, B and C have a common point P . Let circle A intersect circlesB and C at Z and Y , circle B intersect circle C at X. Show that AX, BY and CZ are concurrent.

74.Let ABCD be a cyclic quadrilateral, K, L, M , N midpoints of sides AB, BC, CD, DA respectively. Provethat the orthocenters of triangles AKN , BKL, CLM , DMN are vertices of a parallelogram.

75.In an acute-angled triangle ABC the interior bisector of the angle Aintersects BC at L and intersects thecircumcircle of ABC again at N .From point L perpendiculars are drawn to AB and AC, the feet of theseperpendiculars being K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC haveequal areas.

76.Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of thesegment EB. The tangent line at E to the circle through D, E, and M intersects the lines BC and AC at Fand G, respectively. If

AM

AB= t,

findEG

EF

in terms of t.By mecrazywong

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77.Let M be a point on the side AB of ∆ABC. Let r1, r2 and r be the radii of the inscribed circles of trianglesAMC,BMC and ABC. Let q1, q2 and q be the radii of the escribed circles of the same triangles that lie inthe angle ACB. Prove that

r1

q1· r2

q2=

r

q

78.Let ABCD be a convex quadrilateral with 6 BAC = 6 DAC. Let E be a point on AC, BE and AC intersectat F , DF and BC intersect at G. Show that 6 GAC = 6 EAC.

79.In 4ABC, let A′, B′, C ′ be the midpoints of BC, CA, AB respectively. Let S be the incenter of 4A′B′C ′.Show that S, G, I are collinear and IG : GS = 2 : 1. where G and I are the centroid and incenter of 4ABC.

80.In an acute-angled triangle ABC, let X, Y, Z be points on sides AB, BC and CA respectively. Show thatXY + Y Z + ZX is minimum if and only if 4XY Z is the orthic triangle of 4ABC.

81.In a quadrilateral ABCD, AD//BC, E is a point on side AB, O1 and O2 are the circumcenters of 4AEDand 4BEC respectively. Prove that O1O2 is independent of the position of E.

82.ABC is an isosceles triangle with AB = AC. Suppose that

1. M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB;

2. Q is an arbitrary point on the segment BC different from B and C;

3. E lies on the line AB and F lies on the line AC such that E, Q, F are distinct and collinear.

Prove that OQ is perpendicular to EF if and only if QE = QF .

83.Given a convex quadrilateral has inscribed circle. The tangency points of sides AB,BC, CD,DA areA1, B1, C1, D1 respectively. Let E,F,G,H be the midpoints of A1B1, B1C1, C1D1, D1A1 respectively. Showthat EFGH is a rectangle if and only if A,B,C, D are concyclic.

84.Given a circle O1 with diameter BC. Another circle O2 is drawn through points B and C. Let the tangentline of C to O1 meet O2 at point A. Let AB ∩ O1 = E, CE ∩ O2 = F . H is a point on AF , HE ∩ O1 = G,

BG ∩AC = D. Show thatAH

HF=

AC

CD.

85.Let 4DEF be the orthic triangle of 4ABC. Show that

DE + EF + FD ≤ 12(AB + BC + CA)

86.Let D be a point on side BC of 4ABC. Let O1 and O2 be the circumcenters of 4ABD and 4ADCrespectively. Show that the perpendicular bisector of the median AK passes through the midpoint of O1O2.

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87.Let O be the circumcenter of an acute-angled triangle ABC. Line AO meets side BC at D. Points E andF are on AB and AC respectively such that AEDF is concyclic. Let EG ⊥ BC at G, FH ⊥ BC at H. Showthat GH remains constant.

88.Let ABCD be a cyclic quadrilateral with diameter AC, AC ⊥ BD. F , G are points on lines DA, BAand outside the circumcircle of ABCD, such that BF//DG. Let CH ⊥ CF at H. Prove that B,E, F, H areconcyclic.

89.Given P is a point outside circle O. PA and PB are tangents to circle O, where the tangency points areA and B. Let PO and AB intersect at point Q. A chord CD is drawn through Q. Prove that 4PAB and4PCD have the same incenters.

90.Let ABCD be a cyclic quadrilateral with diameter AB, S the intersection point of AC and BD, T the footof perpendicular from S to AB. Show that TS, AD and BC are concurrent.

91.In a triangle ABC, let X, Y , Z be points on BC, CA, AB such that AX, BY , CZ are concurrent. Thecircumcircle of 4XY Z intersect BC, CA, AB again at X ′, Y ′, Z ′ respectively. Show that AX ′, BY ′, CZ ′ areconcurrent.

92.Let ra, rb, rc be the exradii of 4ABC, r the inradius of 4ABC. Show that

1ra

+1rb

+1rc

=1r

93.Let IA, IB , IC be the excenters of 4ABC outside BC, CA, AB respectively. Show that the circumcenterV of 4IAIBIC , incenter I and circumcenter O of 4ABC are collinear with OI = OV and V IA = 2OA.

94.Let the incircle of 4ABC touch side BC at X. Show that the incircles of 4ABX and 4ACX touch AXat the same point.

95.Two circles Γ1 and Γ2 intersect at M and N . Let l be the common tangent to Γ1 and Γ2 so that M is closerto l than N is. Let l touch Γ1 at A and Γ2 at B. Let the line through M parallel to l meet the circle Γ1 againat C and the circle Γ2 again at D. Lines CA and DB meet at E; lines AN and CD meet at P ; lines BN andCD meet at Q. Show that EP = EQ.

96.Let BC be a diameter of the circle Γ with center O. Let A be a point on Γ such that 0◦ < 6 AOB < 120◦.Let D be the midpoint of the arc AB not containing C. The line through O parallel to DA meets the line ACat J . The perpendicular bisector of OA meets Γ at E and at F . Prove that J is the incenter of the triangleCEF .

97.Let I,G, S, M be the incenter, centroid, cleavance-center and Nagel point of 4ABC. Show that I,G, S, Mare collinear and IG : GS : SM = 2 : 1 : 3.

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98.Show that the circumcircles of the four triangles formed by four straight lines in general position areconcurrent.

99.Given two circles C1 and C2 intersect at S. A,B are points on C2 and C1 such that SA and SB are tangentto C1 and C2 respectively. P,Q are points on C1 and C2 such that PSQ is tangent to the circumcircle of4ABS. Show that SP = SQ.

100.Let P be a point on the circumcircle of 4ABC, H the orthocenter of 4ABC. Show that the midpoint ofHP lies on the nine-point circle of 4ABC and also the Simson line of P relative to 4ABC.

101.(Centroid of quadrilateral) Let P,Q,R, S be the midpoints of sides AB,BC, CD,DA of a quadrilateralABCD. G is the intersection point between PR and QS. Prove that PG = GR and QG = GS.

102.(Anticenter of cyclic quadrilateral) Show that the perpendiculars from the midpoints of the sides to theiropposite sides of a cyclic quadrilateral are concurrent.

103.Show that the circumcenter O, anticenter T and centroid G of a cyclic quadrilateral are collinear withGO = GT .

104.Let E and F be the midpoints of the diagonals AC and BD of a cyclic quadrilateral ABCD. AC and BDintersect at point P . Show that the orthocenter of 4PEF is the anticenter of quadrilateral ABCD.

105.(Isogonal Conjugate) Let X be a point inside 4ABC. Reflect lines AX, BX, and CX about the anglebisectors at A, B, and C respectively. Prove that the three reflected lines are concurrent.

106.Let M,N,P, Q be points on sides AB,BC, CD,DA of quadrilateral ABCD respectively. Show that linesMQ,NP, BD are concurrent if and only if lines MN,PQ,AC are concurrent.

107.ABCDEF is a hexagon inscribed in a circle. Show that the diagonals AD,BE,CF are concurrent if andonly if AB · CD · EF = BC ·DE · FA.

108.Let ABC be a triangle and the incircle, with center I, meet BC at D. Let M be the midpoint of BC.Prove MI bisects AD.

109.ABCD is a convex quadrilateral, where AC and BD intersects at P . M,N are the midpoints of AB andCD respectively. Prove that the lines passing through P,M,N which is perpendicular to AD,BD, AC areconcurrent if and only if A,B, C, D are concyclic.

110.From a point P outside a circle centered at O, draw the two tangents to the circle touching it at A,B. LetM be a point on the segment AB and let C,D be points on the circle with midpoint M . Let the tangents tothe circle at C,D intersect at Q. Show that OQ ⊥ PQ.

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111.Let ABCD be a quadrilateral with inscribed circle. Let the inscribed circle touch AB,BC, CD,DA atpoints E,G, F,H respectively. Prove that AC,BD, EF, GH are concurrent.

112.Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy AB = AD + BC. Thereexists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD andBP = h + BC. Show that:

1√h≥ 1√

AD+

1√BC

.

113.Given a triangle ABC, let I be the center of its inscribed circle. The internal bisectors of the anglesA,B,C meet the opposite sides in A′, B′, C ′ respectively. Prove that

14

<AI ·BI · CI

AA′ ·BB′ · CC ′ ≤827

.

114.Let D be a point inside the acute-angled triangle ABC such that 6 ADB = 6 ACB + 90◦ and AC ·BD =AD ·BC.

(a) Calculate the value of the ratioAB · CD

AC ·BD.

(b) Prove that the tangents at C to the circumcircles of 4ACD and 4BCD are perpendicular.

115.Let R and r be the circumradius and inradius, respectively, of4ABC, and let R′ and r′ be the circumradiusand inradius, respectively, of 4A′B′C ′. Prove that if 6 C = 6 C ′ and Rr′ = R′r, then the triangles are similar.

116.Let ABCD be a convex quadrilateral with 6 BAC = 6 CAD and 6 ABC = 6 ACD. Rays AD and BCmeet at E and rays AB and DC meet at F . Prove that

(a) AB ·DE = BC · CE;

(b) AC2 <12(AD ·AF + AB ·AE)

117.Let ABC be a triangle and P an interior point in ABC. Show that at least one of the angles 6 PAB,6 PBC, 6 PCA is less than or equal to 30◦.

118.In a plane, given two parallel lines k, l and a circle not intersecting k. Let A be a point on k. Let thetangents from A to the circle meet l at points B and C, m the line passing through A and the midpoint ofBC. Prove that when A moves along k, m still passes through a fixed point in the plane.

119.Let A1, A2, A3 be the midpoints of W2W3,W3W1,W1W2 respectively. From Ai (i = 1, 2, 3) drop a per-pendicular to the tangent line to the circumcircle of 4W1W2W3 at Wi. Show that these perpendiculars areconcurrent.

By mecrazywong

11

120.Let P be a point inside 4ABC such that 6 PAB = 6 PBC = 6 PCA = ω. Prove that cotω = cotA +cot B + cot C and ω ≤ 30◦.

121.Let K and N be the points on sides AB and AD of square ABCD, such that AK · AN = 2BK · DN .Segments CK and CN meet the diagonal BD at L and M respectively. Show that K, L, M, N,A are concyclic.

122.Let ABCD be a convex quadrilateral, I1 and I2 be the incenters of 4ABC and 4DBC respectively. Theline I1I2 intersects lines AB and DC at points E and F respectively. Given AB and DC intersect at P , andPE = PF , prove that A,B,C, D are concyclic.

123.Given l is the perimeter of an acute-angled 4ABC which is not an equilateral triangle, P is a variablepoint inside 4ABC, and D,E, F are the projections of P on BC, CA and AB respectively. Prove that2(AF + BD + CE) = l if and only if P is collinear with the incenter and circumcenter of 4ABC.

124.Let D,E, F be the points on sides BC, CA,AB of4ABC respectively, such that DB = DF and DC = DE.Let H be the orthocenter of 4ABC. Show that A,E, F, H are concyclic.

125.Let S be the circumcircle of 4ABC. Reflect S with respect to AB,BC, CA to get three new circles. Provethat the three circles intersect at a common point.

126.Let A′, B′, C ′ be respective midpoints of the arcs BC, CA,AB not containing points A,B,C respectively,of the circumcircle of 4ABC. Let BC, CA,AB meet the pairs of segments C ′A′, A′B′, A′B′, B′C ′, B′C ′, C ′A′

at the points M,N , P,Q, R,S respectively. Prove that MN = PQ = RS if and only if 4ABC is equilateral.

127.In 4ABC, points D and E are chosen on CA such that AB = AD and BE = EC (E lying between Aand D). Let F be the midpoint of arc BC of the circumcircle of 4ABC. Show that B,E,D, F are concyclic.

128.Let the diagonals of a convex quadrilateral ABCD intersect at point O. Let G1, G2 be the centroidsof 4AOB and 4COD respectively, H1,H2 the orthocenters of 4BOC and 4DOA respectively. Show thatG1G2 ⊥ H1H2.

129.Let A1A2...An be a regular polygon inscribed in a circle C with radius R. Let P be an arbitrary point on

C. Show thatn∑

i=1

PA2i = 2nR2.

130.Let O be the circumcenter of 4ABC with AB = AC, D the midpoint of AB and E the centroid of 4ACD.Prove that OE ⊥ CD.

131.Let ABCDEF be an hexagon inscribed in a circle with AB = CD = EF = R, where R is the circumradius.Let P,Q,R be the midpoints of BC, DE, FA respectively. Prove that 4PQR is equilateral.

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132.Let ABCD be a convex quadrilateral such that diagonals AC and BD intersect at right angles, and let Obe their intersection. Prove that the reflections of O across AB,BC, CD,DA are concyclic.

133.In acute-angled triangle ABC, 6 C > 6 B. D is a point on BC such that 6 ADB is obtuse. H is theorthocenter of 4ABD. Point F is in the interior of 4ABC and on the circumcircle of 4ABD. Prove that Fis the orthocenter of 4ABC if and only if HD//CF and H lies on the circumcircle of 4ABC.

134.Prove that if all the sides of an octagon are rational numbers and all its interior angles are equal, then theopposite sides are equal.

135.Let D be a point inside acute-angled triangle ABC. Prove that DA·DB·AB+DB·DC ·BC+DC ·DA·CA ≥AB ·BC · CA, and determine the geometric position of D when the equality holds.

136.Let ABCD be a cyclic quadrilateral, AB,DC and AD,BC intersect at points P and Q respectively. FromQ draw the tangents QE and QF to the circumcircle of ABCD, where the tangency points are E and F . Showthat P,E, F are collinear.

137.Let ABC be a triangle inscribed in a circle and let

la =ma

Ma, lb =

mb

Mb, lc =

mc

Mc,

where ma, mb, mc are the lengths of the angle bisectors (internal to the triangle) and Ma, Mb, Mc are thelengths of the angle bisectors extended until they meet the circle. Prove that

la

sin2 A+

lb

sin2 B+

lc

sin2 C≥ 3,

and that equality holds iff ABC is an equilateral triangle.

138.Let AB be the diameter of a circle O, C a point on segment AB. D is a point on O such that DC = DB.Let a circle centered at point E tangent to O, CD and the circle with diameter AC. Show that EC ⊥ AB.

139.Let the area of a bicentric polygon (i.e. a polygon which has both circumcircle and incircle) be B, the areaof its circumcircle and incircle A and C respectively. Show that 2B < A + C.

140.Let C(I) be a circle centered at the incenter I of4ABC. Points D,E, F are the intersection points betweenC(I) and the perpendiculars from I to BC, CA,AB respectively. Show that AD,BE,CF are concurrent.

141.In a 4ABC, let the perpendicular bisector of BC and the internal bisector 6 A intersect at point P . LetPX ⊥ AB at X, PY ⊥ AC at Y . XY and BC intersect at Z. Show that BZ = ZC.

142. Six points are given on a circle. The orthocenter of the triangle formed by 3 of the points is joined by asegment to the centroid of the triangle formed by other 3 points. Show that the 20 segments obtained in thisway are all concurrent.

143.Let ABC be an acute-angled triangle, D the foot of the perpendicular from A to BC. With AD asa diameter, draw a circle intersecting AB and AC at points E and F respectively. Suppose AD and EFintersect at G, and AD is extended to meet the circumcircle of 4ABC at H. Show that AD2 = AG ·AH.

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144.Let M be the intersection point of the diagonals AC and BD of a convex quadrilateral ABCD. The bisectorof 6 ACD meets BA extended at K. If MA ·MC + MA · CD = MB ·MD, show that 6 BKC = 6 CDB.

145.Let D be a point on side AC of 4ABC, E and F the points on segments BC and BD respectively, suchthat 6 BAE = 6 CAF . Let P,Q be the points on segments BC and BD, satisfying EP//QF//DC. Show that6 BAP = 6 QAC.

146.Let A,B,C be the points on sides B1C1, C1A1, A1B1 of 4A1B1C1 respectively, such that 6 ABC =6 A1B1C1, 6 BCA = 6 B1C1A1, 6 CAB = 6 C1A1B1. Show that the distances between the orthocenters of4ABC, 4A1B1C1 and the circumcenter of 4ABC are the same.

147.Let ABC be a triangle with AB = AC, and points M and N on AB and AC respectively. The lines BNand CM intersect at P . Prove that MN//BC if and only if 6 APM = 6 APN .

148.Let ABC be an acute-angled triangle with angle bisectors BL and CM . Prove that 6 A = 60◦ if and onlyif there exists a point K on BC, with K 6= B,C, such that 4KLM is equilateral.

149.Let the incircle of 4ABC touches sides BC, CA,AB at points D,E, F respectively. Let G be the foot of

perpendicular from D to EF . Show thatFG

EG=

BF

CE.

150.A circle passing through vertices B and C of 4ABC intersects sides AB and AC at C ′ and B′ respectively.Prove that the lines BB′, CC ′,HH ′ are concurrent, where H and H ′ are the orthocenters of 4ABC and4AB′C ′ respectively.

151.In a 4ABC, AB = AC, let D be a point on AB and E a point on AC produced such that DE = AC.Suppose DE meet the circumcircle of 4ABC at T . Let P be a point on AT produced. Prove that PD+PE =AT if and only if P is on the circumcircle of 4ADE.

152.Points D and E lie on the side AB of 4ABC and satisfyAD

DB·AE

EB= (

AC

CB)2. Prove that 6 ACD = 6 BCE.

153.In the exterior of a 4ABC two rectangles ACPQ and BKLC are constructed, such that the areas of theserectangles are equal. Prove that the midpoint of the segment PL, the point C and the circumcenter of the4ABC are collinear.

154.Let ABC be a triangle with CA = CB. A point P lies inside 4ABC, such that 6 PAB = 6 PBC. Thepoint M is the midpoint of AB. Prove that 6 APM + 6 BPC = 180◦.

155.Let ABCD be a cyclic quadrilateral. Let E and F be variable points on the sides AB and CD, respectively,

such thatAE

EB=

CF

FD. Let P be the point on the segment EF such that

PE

PF=

AB

CD. Prove that the ratio

between the areas of triangles APD and BPC does not depend on the choice of E and F .

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156.Given four points A,B, C, D on a circle such that AB is a diameter and CD is not a diameter. Show thatthe line joining the point of intersection of the tangents to the circle at the points C and D with the point ofintersection of the lines AC and BD is perpendicular to the line AB.

157.From a point O 4 rays are drawn, namely OA, OC, OB,OD in anti-clockwise direction, such that 6 AOB =6 COD. Let S be a circle with tangents OA,OB, T a circle with tangents OC, OD. S and T intersect at pointsE and F . Show that 6 AOE = 6 DOF .

158.Let D be a point on side BC of 4ABC such that AD > BC and let E be a point on side AC such thatAE

EC=

BD

AD −BC. Prove that AD > BE.

159.Let ABC be a triangle and D a point on line BC such that B is lying between D and C and also DB = AB.Let M be the midpoint of the segment AC. Denote P = DM ∩ ` where ` is the bisector of 6 ABC. Prove that6 BAP = 6 BCA.

160.Let M , N and P be points of intersection of the incircle of 4ABC with sides AB, BC and CA respectively.Prove that the orthocenter of 4MNP , the incenter of 4ABC and the circumcenter of 4ABC are collinear.

161.Let A,B,C, D be four points on a circle Γ with center O. The chords AC and BD intersect at a point Pinside the circle Γ. If the tangent at P of the circumcircle of 4PCD passes through O, show that (a)the area

of 4ABP is12R2 sin 2θ; (b) 6 APB ≤ 2θ.

162.In 4ABC, let P be a point on its circumcircle. Suppose that H,A′, B′, C ′ are respectively the orthocentersof 4ABC, 4PBC, 4APC, 4ABP . Prove that 4ABC ∼= 4A′B′C ′ and HP, AA′, BB′, CC ′ are concurrent.

163.Two circles intersect at points A and B. Through the point B a straight line is drawn, intersecting thefirst circle at K and the second circle at M . A line parallel to AM is tangent to the first circle at Q. The lineAQ intersects the second circle again at R. Prove that (a)the tangent to the second circle at R is parallel toAK; (b)these two tangents are concurrent with KM .

164.A triangle ABC is given. A circle Γ passes through vertex A and is tangent to side BC at point P . Thecircle Γ intersects sides AB and AC at points M and N , respectively. Prove that (minor) arcs MP and NPare equal if and only if Γ is tangent to the circumcircle of 4ABC at A.

165.ABCD is a convex quadrilateral such that AB and CD are not parallel. The circle through A,B touchesCD at P , and a circle through C,D touches AB at Q. These two circles intersect at E,F . Show that AD//BCif and only if EF bisects PQ.

166.Let D,E, F be the feet of the angle bisectors of angles A,B, C, respectively, of 4ABC, and let Ka,Kb,Kc

be the points of contact of the tangents to the incircle of 4ABC through D,E, F (that is, tangent linesnot containing sides of triangle). Prove that the line joining Ka,Kb,Kc to the midpoints of BC, CA,AB,respectively, pass through a single point on the incircle of 4ABC.

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167.Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side ABso that both 4ABP and 4ACQ are acute. Let R be the orthocenter of 4ABP and S be the orthocenter of4ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of 6 CBP and6 BCQ such that 4TRS is equilateral.

168.Given two circles intersect at points A and B. A line passes through point A and intersects the two circlesat points C and D. Let M,N be the midpoints of arcs BC, BD (the part not containing A) respectively, Kthe midpoint of CD. Show that 6 MKN = 90◦.

169.Let two equal regular n-gons S and T be located in the plane such that their intersection is a 2n-gon(n ≥ 3). The sides of the polygon S are coloured in red and the sides of T in blue. Prove that the sum of thelengths of the blue sides of the polygon S ∩ T is equal to the sum of the lengths of its red sides.

170.Let ABC be a triangle with AB 6= AC. Let C1 be the point such that AC1 = BC1 and BC1 ⊥ BC, B1

the point such that AB1 = CB1 and CB1 ⊥ CB. If D is the point on line BC such that AD is tangent to thecircumcircle of ABC, prove that D,B1, C1 are collinear.

171.Let ABC be a triangle, D a point on BC and ω the circumcircle of 4ABC. Show that the circles tangentto ω, AD, BD and to ω, AD, DC, respectively, are tangent to each other if and only if 6 BAD = 6 CAD.

172.Let ABCD be a quadrilateral with AC = BD. Construct 4 isoceles triangles AHB,BMC,CPD,DNAoutside qaadrilateral ABCD, such that AH = HB, BM = MC,CP = PD,DN = NA and 4AHB ∼ 4CDP ,4BMC ∼ 4DNA. Prove that HP ⊥ MN .

173.(Orthologic triangles)Let ABC and DEF be two triangles. Show that the perpendiculars through D,E, Fto BC, CA,AB, respectively, are concurrent if and only if the perpendiculars through A,B, C to EF,FD,DE,respectively, are concurrent.

174.(Pierre-Leon Anne’s Theorem)Prove that in any quadrilateral ABCD which is not a parallelogram, thelocus of the points O such that [AOB] + [COD] = [BOC] + [AOD] is Newton’s Line of the quadrilateral. TheNewton’s Line is the line joining the midpoints of the diagonals of a quadrilateral which is not a parallelogram.

175.(Newton’s Theorem)Prove that the center of the circle inscribed in a quadrilateral lies on the line joiningthe midpoints of the diagonals.

176.ABC is a triangle with angle bisectors AD and BE. The lines AD,BE meet the line through C parallelto AB at points F and G respectively. Show that DF = EG if and only if CA = CB.

177.In the interior of a cyclic quadrilateral ABCD, a point P is given such that 6 BPC = 6 BAP + 6 PDC.Denote by E,F,G the feet of the perpendicular from P to AB,AD,DC respectively. Show that 4EFG ∼4BPC.

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178.Let the inscribed circle of 4ABC be tangent to sides AB,BC at points E and F respectively. Let theangle bisector of 6 A intersect EF at K. Prove that 6 CKA = 90◦.

179.Let K, L, M be the points on sides AB,BC, CA of a 4ABC such thatAK

KB=

BL

LC=

CM

MA. Show that

4ABC and 4KLM have a common orthocenter if and only if 4ABC is equilateral.

180.Let ABC be a triangle. E,F are the points on side BC such that the semicircle with diameter EF touchesAB and AC at points Q and P respectively. Show that the intersection between EP and FQ lies on thealtitude from A to BC.

181.(Generalization of Menelaus’ Theorem)Let A,B, C be three points on one line, P a point not on this line.Let A′, B′, C ′ be three arbitrary points on the lines AP,BP,CP respectively. Prove that A′, B′, C ′ are collinear

if and only if BC · AA′

A′P+ CA · BB′

B′P+ AB · CC ′

C ′P= 0, where all the segments here are directed segments.

182.Let ABC be a triangle with altitudes AD,BE, CF and orthocenter H. M is the midpoint of BC. Letlines AM,HM meet E and F at points X, Y respectively. Show that 6 XDE = 6 Y DF .

183.Let ABCD (AB 6= AD) be a quadrilateral with incircle centered at O. Let OA and OC meet the incircleat points E and F respectively. Prove that 6 DAB = 6 BCD if and only if EF ⊥ BD.

184.Let ABCD be a rectangle with area 2, P a point on side CD. Let the incircle of 4PAB tangent to ABat point Q. Show that when PA ·PB attains its minimum value, AB ≥ 2BC, and find the value of AQ ·BQ.

185.Let O be the circumcenter of 4ABC, P a point inside 4AOB. Let D,E, F be the projections of P onBC, CA,AB respectively. Show that, the parallelogram with FE,FD as the adjacent sides lies in the interiorof 4ABC.

186.Let ABCD be a quadrilateral such that AB + CD = BC + DA. Let the incircles of 4ABC and 4ABDtangent to AC,BC, AD, BD at points E,F, G,H respectively. Show that E,F,G,H are concylic.

187.Let ABC be a triangle. Let M and N be the points in which the median and the angle bisector, respectivelyat A meet the side BC. Let Q and P be the points in which the perpendicular at N to NA meets MA andBA, respectively, and O the point in which the perendicular at P to BA meets AN produced. Prove thatQO ⊥ BC.

188.Let ABC be an acute-angled triangle. Let D,E, F be the feet of the altitudes of 4ABC from the verticesA,B,C respectively. Let one of the tangents from D to the circle with diameter AH meet AC at point E′,whereas the other one meets AB at point F ′. Prove that AD,BE′, CF ′ are concurrent.

189.Let ABC be a triangle, A1, B1, C1 the midpoints of sides BC, CA,AB, respectively, and A2, B2, C2 themidpoints of the altitudes from A,B,C respectively. Show that A1A2, B1B2, C1C2 are concurrent.

190.Let ABCDE be a convex pentagon such that 6 ABC = 6 ADE and 6 AEC = 6 ADB. Show that 6 BAC =6 DAE.

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191.Let A′, B′, C ′ be the points on sides BC, CA,AB of 4ABC such that AA′, BB′, CC ′ concur at point M .MB and A′C ′ intersect at point P , MC and A′B′ intersect at point Q. Prove that 6 MAB = 6 MAC if andonly if 6 MAP = 6 MAQ.

192.Let P1, P2, P3, P4 be four points on a circle, and let I1, I2, I3, I4 be the incenters of 4P2P3P4,4P1P3P4,4P1P2P4,4P1P2P3. Prove that I1, I2, I3, I4 are the vertices of a rectangle.

193.Let ABCD be a quadrilateral AB = BC = CD = DA. Let MN and PQ be two segments perpendicular

to the diagonal BD and such that the distance between them is d >BD

2, with M ∈ AD, N ∈ DC, P ∈ AB,

and Q ∈ BC. Show that the perimeter of hexagon AMNCQP does not depend on the position of MN andPQ so long as the distance between them remains constant.

194.Let ABCD be a quadrilateral such that all sides have equal length and 6 ABC = 60◦. Let l be a line passingthrough D and not intersecting the quadrilateral (except at D). Let E and F be the points of intersection of lwith AB and BC respectively. Let M be the point of intersection of CE and AF . Prove that CD2 = CM ·CE.

195.Let ABC be a triangle and D,E, F the points on BC, CA,AB respectively such that the cevians AD,BE,CF are concurrent. Let M,N,P be points on EF,FD,DE respectively. Show that AM,BN,CP are concur-rent if and only if DM, EN, FP are concurrent.

196.Let x be an integer such that there exists two non-congruent traingles with the same area x. Find theminimum value of x.

197.Let ABCDEF be a convex hexagon such that AB is parallel to DE, BC is parallel to EF , and CD isparallel to FA. Let RA, RC , RE denote the circumradii of triangles FAB, BCD, DEF , respectively, and let Pdenote the perimeter of the hexagon. Prove that

RA + RC + RE ≥ P

2.

198.In the plane let C be a circle, L a line tangent to the circle C and M a point on L. Find the locus of allpoints P with the following property: there exists two points Q,R on L such that M is the midpoint of QRand C is the inscribed circle of 4PQR.

199.In an acute-angled triangle ABC the internal bisector of angle A meets the circumcircle of the triangleagain at A1. Points B1 and C1 are defined similarly. Let A0 be the point of intersection of the line AA1 withthe external bisectors of angles B and C. Points B0 and C0 are defined similarly. Prove that:

(i) The area of the triangle A0B0C0 is twice the area of the hexagon AC1BA1CB1.

(ii) The area of the triangle A0B0C0 is at least four times the area of the triangle ABC.

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200.Let ABCD be a quadrilateral with incircle centered at O. Prove that

OA ·OC + OB ·OD =√

AB ·BC · CD ·DA

201.Let F be the Fermat’s Point of 4ABC, D = BF ∩AC and E = CF ∩AB. Prove that AB +AC ≥ 4DE.

202.Let P be an interior point of 4ABC. Suppose the lines BP and CP meet AC and AB at points E and Frespectively. Let D be the point where AP meets EF and K the foot of perpendicular from D to BC. Showthat DK bisects 6 EKF .

203.Let I be the incenter of 4ABC. Let a circle which is tangent to the the circumcircle of 4ABC, AB,BC,touch the circumcircle at point D. Show that DI bisects 6 ADC.

204.Let the incircle of4ABC touch sides BC, CA,AB at points D,E, F respectively. Let the external bisectorsof 6 A, 6 B, 6 C meet the circumcircle of 4ABC at points D′, E′, F ′ respectively. Show that DD′, EE′, FF ′ areconcurrent.

205.Let D,E, F be the midpoints of sides BC, CA,AB of 4ABC .The incircles of triangles AEF,BFD, CDEare tangent to EF,FD,DE at points P,Q,R respectively. Prove that PD,QE, RF are concurrent at theincenter of 4ABC.

206.Let D,E, F be the points on sides BC, CA,AB of4ABC such that the incircles of4AEF,4BFD,4CDEhave the same inradii r. Let r0 and R be the inradii of the incircles of 4DEF and 4ABC respectively. Provethat r + r0 = R.

207.Let 4ABC be inscribed in a circle O. Let D be a point on ray BC such that AD is tangent to circle O.P is a point on ray DA. A line through P cuts circle O at points Q,T , segment CD at point U and AB andAC at point R,S respectively. Prove that if QR = ST , then PQ = UT .

208.A circle meets the three sides BC, CA,AB of a triangle ABC at points D1, D2;E1, E2;F1, F2 respectively.Furthermore, line segments D1E1 and D2F2 intersect at point L, line segments E1F1 and E2D2 intersect atpoint M , line segments F1D1 and F2E2 intersect at point N . Prove that the lines AL, BM, CN are concurrent.

209.Let a, b, c be the three sides of 4ABC, b < c. AD is the internal angle bisector of 6 A with D on BC.

(1) Find the necessary and sufficient condition(in terms of angles A,B,C) such that there exist points E,Fon AB,AC(excluding the endpoints) satisfying BE = CF and 6 BDE = 6 CDF .

(2) Under the condition that E and F exist, express the length of BE in terms of a, b, c.

210.For any 4 distinct points P1, P2, P3, P4 in the plane, find the minimum value of the ratio

∑1≤i≤j≤4

PiPj

min1≤i≤j≤4

PiPj.

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211.Let D be a point on side AB of a equilateral triangle ABC such that AD =1n

AB (n ≥ 3). P1, P2, ..., Pn−1

are points on BC which divide BC into n equal parts. Show that 6 DP1A + 6 DP2A + ... + 6 DPn−1A = 30◦.

212.Prove that for any acute-angled triangle, there exists a point P such that the feet of perpendiculars fromP to the three sides of the triangle are vertices of a equilateral triangle.

213.There are 5 points in a rectangle(including its boundary) with area 1, no three of them are in the same

line. Find the minimum number of triangles with the area not more than14, vertex of which are three of the

five points.

214.Let I,H be the incenter and orthocenter of an acute-angled 4ABC, B1, C1 the midpoints of sides AC,AB.It is known that ray B1I meets side AB at point B2 (B2 6= B), ray C1I meets side AC produced at point C2.B2C2 and BC intersect at K. Let A1 be the circumcenter of 4BHC. Prove that A, I, A1 are collinear if andonly if the areas of 4BKB2 and 4CKC2 equal.

215.Let H be the orthocenter of 4ABC. Using pure geometry, show that BC2 + AH2 = 4R2, where R is thecircumradius of the circumcircle of 4ABC.

216.Let A,B,C, D be four distinct points arranged in order on a circle. The tangent to the circle at A meets theray CB at K and the tangent to the circle at B meets the ray DA at H. Suppose BK = BC and AH = AD.Prove that the quadrilateral ABCD is a trapezium.

217.Let ω be the incircle of triangle ABC. Suppose that there exists a circle passing through B and C andtangent to ω at A′. Suppose the similar points B′, C ′ exist. Prove that the lines AA′, BB′, CC ′ are concurrent.

218.(Apollonian Circle)Let A,B be two distinct points in the plane. Find the locus of point P such thatAP

PB= r, where 0 < r < 1.

219.Let ABC be a triangle and let P be a point in its interior. Lines PA, PB, PC intersect sides BC, CA,AB at D, E, F , respectively. Prove that

[PAF ] + [PBD] + [PCE] =12[ABC]

if and only if P lies on at least one of the medians of triangle ABC. (Here [XY Z] denotes the area of triangleXY Z.)

220.Let quadrilateral ABCD be circumscribed in a circle with center I(I /∈ AC). Diagonals AC and BDintersect at point E. A line perpendicular to BD and passing through E intersects lines IA and IC at pointsP and Q respectively. Prove that PE = EQ.

221.Let Γ1 and Γ2 be two circles intersecting at P and Q. The common tangent, closer to P , of Γ1 and Γ2

touches Γ1 at A and Γ2 at B. The tangent of Γ1 at P meets Γ2 at C, which is different from P , and theextension of AP meets BC at R. Prove that the circumcircle of 4PQR is tangent to BP and BR.

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222.Given a convex quadrilateral ABCD. Prove that there exists a point N , such that both 4BNC and4DNA are equilateral triangle if and only if there exists a point M , such that 4AMB and 4CMD areisosceles and 6 AMB = 6 CMD = 120◦.

223.Let E be the intersection point of the diagonals of a convex quadrilateral ABCD. Denote F, F1, F2 bythe areas of ABCD,4ABE,4CDE respectively. Show that

√F1 +

√F2 ≤

√F and determine when equality

holds.

224.D,E, F are points on sides BC, CA,AB of an acute-angled 4ABC respectively (not the endpoints),satisfying EF//BC. D1 is a point on side BC (different from B,D,C). E1, F1 are points on sides AC,ABrespectively such that D1E1//DE, D1F1//DF . Construct a 4PBC above BC (on the same side of A) suchthat 4PBC ∼ 4DEF . Prove that EF,E1F1, PD1 are concurrent.

225.Let D be a point on side BC of triangle ABC and P a point on segment AD. A line through D intersectssegments AB, PB at M , E respectively, also intersects the extensions of segments AC, PC at F , N respectively.If DE = DF , prove that DM = DN .

226.Let D be a point on side BC of 4ABC. F is a point in 4ABC and on the circle passing through pointsA, D and C. The circle passing through points B, D and F intersects side AB at point E. Prove thatCD · EF + DF ·AE = BD ·AF if and only if AB = AC.

227.Let BM, CN be the medians of 4ABC. Lines BM and CN intersect the circumcircle of 4ABC atpoints P and Q respectively. The tangents at P and Q to the circle intersect AC and AB at points X and Yrespectively. Prove that OG ⊥ XY , where O is the circumcenter and G is the centroid.

228.Let ABCD be a cyclic quadrilateral and AD is not parallel to BC. E,F are points on CD, G, H are thecircumcenters of 4BCE and 4ADF respectively. Prove that AB,CD,GH are concurrent or parallel to eachother if and only if A,B, E, F are concyclic.

229.Given that circles S1 and S2 intersect at two points A and B. Through A draw a line intersecting S1, S2

at points C,D respectively. M,N,K are points on CD,BC,BD such that MN//BD,MK//BC. E,F arepoints on S1 and S2 respectively such that EN ⊥ BC, FK ⊥ BD. Show that 6 EMF = 90◦.

230.Let AD be the angle bisector of an acute-angled 4ABC. E,F are points on AC,AB respectively suchthat DE ⊥ AC,DF ⊥ AB. Let H be the intersection point of BE, CF . The circumcircle of 4AFH and BEintersect at point G. Prove that BG,GE, BF are sides of a right-angled triangle.

231.Let ABCD be a convex quadrilateral such that 6 DAB = 6 ABC = 6 BCD. Let O,H be the circumcenter,orthocenter of 4ABC respectively. Prove that O,H,D are collinear.

232.Let C1 and C2 be two circles intersecting at points A and B. let l1 and l2 be two lines passing through Bintersecting C1 at points D and E and C2 at points F and G. If D,E, F, G lie on a circle with center O, provethat 6 BAO = 90◦.

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233.Let M be the midpoint of side BC of an acute-angle 4ABC. ω is a circle with diameter AM . Let D andE be the intersection points of ω and AB,BC respectively. P is the intersection point of the tangents throughD and E to ω. Prove that PB = PC.

234.Let H and G be the orthocenter and centroid of a non-isosceles acute-angled 4ABC respectively. Show

that 6 AGH = 90◦ if and only if1

[HAB]+

1[HAC]

=2

[HBC].

Problems Source: Except No. 7(b) and 183, which were created by me, all the problems in the Geometry Prob-lems Collection are extracted from MathDB(www.mathdb.org), MathLinks(www.mathlinks.ro), Math Excaliburand various Math Olympiads(for instance, IMO, APMO, CMO, etc.)

Readers are encouraged to send solutions to the problems to samcrazywong@yahoo.com.hk in LaTeX format.Your solutions will be added to the latest version of the solution file if it’s correct and not yet in the file. Ifthat problem was solved, then please feel free to send a different solution. However, solutions essentially thesame as that in the file will NOT be added.

By mecrazywong

-The End-

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