Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles Outline Perpendicular...
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Transcript of Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles Outline Perpendicular...
![Page 1: Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles Outline Perpendicular bisector, circumcentre and orthocenter Bisectors of angles.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649d0a5503460f949dcd34/html5/thumbnails/1.jpg)
Chapter 2 :Circumcenter, Orthocenter, incenter, and centroid of triangles
Outline•Perpendicular bisector ,
circumcentre and orthocenter •Bisectors of angles and the incentre
•Medians and centroid
![Page 2: Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles Outline Perpendicular bisector, circumcentre and orthocenter Bisectors of angles.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649d0a5503460f949dcd34/html5/thumbnails/2.jpg)
2.1 Perpendicular bisector, Circumcenter and orthocenter of a triangle
Definition 1 The perpendicular bisector
of a line segment is a line perpendicular
to the line segment at its midpoint.
C
A B
DCD is a perpendicular bisector of AB if
(i) AC=BC;
(ii) DCA = DCB= o90
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In-Class-Activity 1
(1) If P is a point on the perpendicular bisector of AB, what is the relationship between PA and PB?
(2) Make a conjecture from the observation in (1). Prove the conjecture.
(3) What is the converse of the conjecture in (2).
Can you prove it?
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Theorem 1 The perpendicular bisectors of the three sides of a triangle meet at a point
which is equally distant from the vertices of the triangle.
The point of intersection of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle.
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O
A
C
B
GF
HD E
M
DG, MH and EF are the perpendicular bisectors of the sides AB,AC and BC respectively
(i) DG, MH and EF meet at a point O;
(ii) OA=OB=OC;
(iii) O is the circumcenter of triangle ABC.
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Proof of Theorem 1
Given in ABC that DG, EF and MH are the perpendicular bisectors of sides AB, BC and CA respectively.
To prove that
DG,EF and MH meet at a point O,
and AO=BO=CO.
Plan: Let DG and EF meet at a point O. Then show that OM is perpendicular to AC.
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Proof
1.Let DG and EF meet at O
2. Connect M and O.
We show MO is
perpendicular to side AC
3. Connect AO, BO and CO.
(If they don’t meet, then DG//EF, so AB//BC, impossible)
M
O
ED
CA
B
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M
O
ED
CA
B
4. AO=BO, BO=CO
5. AO=CO
6. MO=MO
7. AM=CM
8.
9.
10
(O is on the perpendicular bisects of AB and BC)
( By 4 )
(Same segment )
( M is the midpoint )
(S.S.S)
(Corresponding angles
(By 9 and )
COMAOM
OMCOMA 090 OMCOMA 0180 OMCOMA
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11.OM is the perpendicular (Two conditions satisfied)
bisector of side AC.
12. The three perpendicular
bisector meet at point O.
13.O is equally distant from ( by 4)
vertices A,B and C.
M
O
ED
CA
B
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Remark 1 ( A method of proving that three lines meet at a point )
In order to prove three lines meet at one point, we can
(i) first name the meet point of two of the lines;
(ii) then construct a line through the meet point;
(iii) last prove the constructed line coincides with the third line.
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In-Class-Exercise 1
Prove Theorem 1 for obtuse triangles.
Draw the figure and give the outline of the
proof
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Remark 2 The circumcenter of a triangle is equally distant from the three vertices.
The circle whose center is the circumcenter of a triangle and whose radius is the distance from the circumcenter to a vertex is called the
circumscribed circle
of the triangle.
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In-Class-Activity
(1) Give the definition of parallelograms
(2) List as many as possible conditions for a quadrilateral to be a parallelogram.
(3) List any other properties of parallelogram which are not listed in (2).
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(1) Definition A parallelogram is a quadrilateral with its opposite sides parallel ABCD
(2) Conditions • The opposite sides equal
• Opposite angles equal
• The diagonals bisect each other
• Two opposite side parallel and equal
(3)
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Theorem 2
The three altitudes of a triangle meet at a point.
FD
EAB
C
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Given triangle ABC with altitudes AD, BE and CF.
To prove that AD, BE and CF meet at a point.
Plan is to construct another larger triangle A’B’C’
such that AD, BE and CF are the perpendicular bisectors
of the sides of A’B’C’. Then apply Theorem 1.
j
C'
B' A'
F
E
D
C
A B
![Page 17: Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles Outline Perpendicular bisector, circumcentre and orthocenter Bisectors of angles.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649d0a5503460f949dcd34/html5/thumbnails/17.jpg)
Proof (Brief)
Construct triangle A’B’C’ such that A’B’//AB, A’C’//AC, B’C’//BC1. AB’CB is a parallelogram.2. B’C=AB.3. Similarly CA’=AB.4. CE is the perpendicular bisector of A’B’C’ of side B’A’.5. Similarly BF and AD are perpendicular bisectors of sides
of A’B’C’.
6. So AD, BF and CE meet at a point (by Theorem 1)
j
C'
B' A'
F
E
D
C
A B
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The point of intersection of the three altitudes of a triangle is called the
orthocenter
of the triangle.
FD
EAB
C
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2.2 Angle bisectors , the incenter of a triangle
Angle bisector:
ABD= DBC
In-Class-Exercise 2
(1) Show that if P is a point on the bisector of then the distance from P to AB equals the distance
from P to CB.
(2) Is the converse of the statement in (1) also true?
B C
A
D
ABC
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Lemma 1 If AD and BE are the bisectors of the angles
A and B of ABC, then AD and BE intersect at a point.
Proof Suppose they do not meet. 1. A+ B+ C=180 ( Property of triangles)
2. Then AD// BE. ( Definition of parallel lines)
3. DAB+ EBA=180 ( interior angles on same side )
4. ( AD and BE are bisectors )
BA
EBADAB 22oo 3601802
A
C
B
DE
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5.This contradicts that
The contradiction shows that the two angle bisectors must meet at a point.
oCBABA 180
Proof by contradiction ( Indirect proof)
To prove a statement by contradiction,
we first assume the statement is false,
then deduce two statements contradicting to each other.
Thus the original statement must be true.
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Theorem 3 The bisectors of the three angles of a triangle
meet at a point that is equally distant from the three side
of the triangle.
O
B C
A
F
D
E
The point of intersection of angle bisectors of a triangle is called the incenter of the triangle
[Read and complete the proof ]
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Remark Suppose r is the distance from the incenter to a side of a triangle. Then there is a circle whose center is the incenter and whose radius is r.
This circle tangents to the three sides
and is called the
inscribed circle ( or incircle) of the triangle.
H
G
K
O
B C
A
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Example 1 The sum of the distance from any interior point of an equilateral triangle to the sides of the triangle is constant.
D
F
H
G
A
BC
E
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Proof
1.
2.
3. AB=AC=BC (ABC is equilateral )
4.
5. ( by 1 and 4)
6.
is a constant.
))((21 BCADABCArea
))(())(())(( 21
21
21 EHBCEGACEFAB
CEBAreaAECAreaBEAAreaABCArea
))((21 EHEGEFBCABCArea EHEGEFAD
EHEGEF
D
F
H
G
A
BC
E
![Page 26: Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles Outline Perpendicular bisector, circumcentre and orthocenter Bisectors of angles.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649d0a5503460f949dcd34/html5/thumbnails/26.jpg)
In-Class-Activity
(1) State the converse of the conclusion proved in Example 1.
(2) Is the converse also true?
(3) Is the conclusion of Example 1 true for points outside the triangle?
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2.3 Medians and centroid of a triangle
A median of a triangle is a line drawn from any vertex to the mid-point of the opposite side.
Lemma 2 Any two medians of a triangle meet at a point.
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Theorem 3 The three medians of a triangle meet at a point which is two third of the distance from each vertex to the mid-point of the opposite side.
The point of intersection of the three medians of a triangle is called the centroid of the triangle
OF E
D
C
BA
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Proof (Outline)• Let two median AD and BE meet at O.
• Show
• If CE and AE meet at O’, then
So O is the same as O’
• All medians pass through O.
[ Read the proof ]
ADAO 32
ADAO 32'
O
E
C
BA
D
O'
F
C
BA
D
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Example 2 Let line XYZ be parallel to side BC and pass
through the centroid O of .
BX, AY and CZ are perpendicular to XYZ.
Prove: AY=BX+CZ.
Y
ZX O
B C
A
ABC
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.
E
Y
W
ZX O
B
A
C
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Question
Is the converse of the conclusion in
Example 2 also true?
How to prove it?
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• Summary
• The perpendicular bisectors of a triangle meet at a point---circumcenter, which is equally distant from the three vertices and is the center of the circle outscribing the
triangle. • The three altitudes of a triangle meet at a point---
orthocenter .• The angle bisectors of a triangle meet at a point---
incenter, which is equally distant from the three sides and is the center of the circle inscribed the triangle.
• The three medians of a triangle meet at a point ---centroid. Physically, centroid is the center of mass of the triangle with uniform density.
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Key terms
Perpendicular bisector
Angle bisector
Altitude
Median
Circumcenter
Orthocenter
Incenter
Centroid
Circumscribed circle
Incircle
![Page 35: Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles Outline Perpendicular bisector, circumcentre and orthocenter Bisectors of angles.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649d0a5503460f949dcd34/html5/thumbnails/35.jpg)
Please submit the solutions of 4 problems in Tutorial 2
next time.
THANK YOU
Zhao Dongsheng
MME/NIE
Tel: 67903893
E-mail: [email protected]