CYCLIC POLYGONSBy Greg Wood

Introduction

Chapter 13 covers the theorems dealing with cyclic polygons, special line segments in triangles, and inscribed & circumscribed polygons.

Also, it covers the relationships between those special line segments.

Definition of cyclic

A polygon is cyclic if there exists a circle that contains all of its points.

Cyclic Triangles

Every triangle is cyclic. If you choose any three points on a

circle, a triangle can always be made from those three points. Since its vertices are contained on the circle, it is cyclic.

Corollary to Theorem 68

The perpendicular bisectors of the sides of a triangle are concurrent.

The perpendicular bisectors of a triangle always meet at the center of the circle drawn around it.

The center is equidistant from all three vertices of the triangle.

Inscribed and circumscribed polygons

A polygon is inscribed in a circle if each vertex of the polygon lies on the circle.

The circle is circumscribed about the polygon. A circle is inscribed in a polygon if each side of

the polygon is tangent to the circle, and the polygon is circumscribed about the circle.

The circle is called the incircle of the polygon, and its center is called the incenter of the polygon.

Circumscribing a circle about a triangle First, you construct the perpendicular

bisector of one side of the triangle. Then, you construct the perpendicular

bisector of another side. Where they cross is the center of the

circumscribed circle. Place the compass on the center point,

adjust its length to reach any vertice , and draw a circle with that radius length.

Demonstration

Cyclic Quadrilaterals

A quadrilateral is cyclic if a pair of its opposite angles are supplementary.

This is because the angles are inscribed on the circle. If the opposite angles are equal to 180 degrees, then the arcs they are inscribed onto equal 360 degrees.

Triangles and Incircles

Every triangle has an incircle. The angles bisectors of a triangle are congruent.

They meet at the center (incenter) or the inscribed circle.

The incenter is equidistant from all three sides of the triangle.

Inscribing a circle in a triangle First, you bisect two angles. Where they cross is the center of the

inscribed circle. Construct a perpendicular line from

the center point to where the circle touches any side of the triangle. That is the radius of the incircle.

Draw a circle based on that radius. Demonstration

The Centroid of a Triangle A median of a triangle is a line segment that

joins a vertex to the midpoint of the opposite side.

The medians of a triangle are concurrent, and they intersect at the centroid

The centroid also divides the medians in a ratio of 2:1

The orthocenter of a triangle The lines containing the altitudes of a triangle are concurrent,

and they meet at the orthocenter. The orthocenter is not always inside the triangle. It depends

whether the triangle is obtuse (outside), acute (inside), or right (on the vertex of the right angle.

Ceva’s Theorem

A cevian of a triangle is a line segment that joins a vertex of the triangle to a point on the opposite side.

Perpendicular bisectors are NOT cevians.

Three cevians, AY, BZ, and CX, of triangle ABC are concurrent iff

AXXB*BYYC*CZZA

1

2

16 12

126

3

216*126*123

288288

1

Napoleon’s Thereom

Napoleon’s theorem states that if equilateral triangles are constructed inward or outward on every side of a triangle, their centers form another equilateral triangle.

Summary

The chapter focused on special points and lines in triangles, and circles inscribed or circumscribed about polygons.

The perpendicular bisectors of a triangle meet at the center of its circumscribed circle.

The angle bisectors of a triangle meet at the center of its inscribed circle, the incenter.

The medians, lines that connect a vertex to the midpoint of the opposite side, meet at the centroid of a triangle.

The altitudes meet at the orthocenter of a triangle. Also, quadrilaterals are cyclic iff their opposite angles are

supplementary. Cevians of a triangle are line segments that join a vertex

to any point on the opposite side.