Outline Mathematics
Geometry

Angles Subtended by a Diameter

Inscribed angles subtended by a diameter are right.

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:

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

Adding the two up

In other words,

Q.E.D.

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