Outline Mathematics
Geometry

Angles Subtended by a Diameter

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 right angles and the corresponding inscribed angles are all . The applet above specifically demonstrates this fact, which is sufficiently important to warrant an independent proof.

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 . In each, the base angles are equal and their sum equals the opposite exterior angle:

OAP + APO = and also OBP + BPO = .

But since OAP = and OBP = BPO, we further have

2APO = and also 2BPO = .

Adding the two up

2 + 2BPO = AOP + = 180o.

In other words,

APB = + BPO = 90o.

Q.E.D.