Angles Subtended by a Diameter

Outline Mathematics
Geometry

Now see how this works:

Created with GeoGebra

This is a known and a very useful property of inscribed angles that they measure half the central angle subtended by the same arc, or, which is the same, by the same chord. When a chord is a diameter, the central angles measures two,two,three,four right angles and the corresponding inscribed angles are all right,acute,right,obtuse. The applet above specifically demonstrates this fact, which is sufficiently important to warrant an independent proof. (The statement is often referred to as Thales' theorem.)

Let P be a point on a circle with diameter AB and center O. So that OA = OB = OP, as three radii of the same circle. This makes triangles AOP and BOP isosceles,isosceles,scalene,equal. In each, the base angles are equal and their sum equals the opposite exterior angle:

∠OAP + ∠APO = ∠BOP,AOB,APO,BOP,AOP and also ∠OBP + ∠BPO = ∠AOP,AOB,APO,BOP,AOP.

But since ∠OAP = ∠APO,AOB,APO,BOP,AOP and ∠OBP = ∠BPO, we further have

2∠APO = ∠BOP,AOB,APO,BOP,AOP and also 2∠BPO = ∠AOP,AOB,APO,BOP,AOP.

Adding the two up

2∠APO,AOB,APO,BOP,AOP + 2∠BPO = ∠AOP + ∠BOP,AOB,APO,BOP,AOP = 180°.

In other words,

∠APB = ∠APO,AOB,APO,BOP,AOP + ∠BPO = 90°.

Q.E.D.

Related material Read more...
	Angles in Circle
	Inscribed Angles
	Inscribed and Central Angles in a Circle
	Munching on Inscribed Angles
	Sangaku with Angle between a Tangent and a Chord
	Secant Angles in a Circle
	Secant Angles in a Circle II
	Thales' theorem

|Up| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny