# Circle Theorems

• Nov 10th 2008, 10:41 AM
bearej50
Circle Theorems
Given: Isosceles Triangle ABC is inscribed in a circle with base BC.
Prove: If P is any poin on minor arc BC, then ray PA bisects angle BPC.
• Nov 10th 2008, 12:58 PM
Soroban
Hello, bearej50!

Quote:

Given: Isosceles Triangle $ABC$ is inscribed in a circle with base $BC.$

Prove: If $P$ is any point on minor arc $BC$, then ray $PA$ bisects $\angle BPC.$

Code:

                A               * o *           *    / \    *         *    /  \    *       *    /  o \    *             /      \       *    /        \    *       *  /      o  \  *       *  /            \  *         /              \       B*- - - - - -o- - -*C         *              *           *          *               * * * o                     P

Draw chords $PB$ and $PC$.

Inscribed angle $APB$ is measured by $\tfrac{1}{2}\,\text{arc}(AB).$
Inscribed angle $APC$ is measured by $\tfrac{1}{2}\,\text{arc}(AC).$

Since $AB = AC$ .
(the triangle is isosceles),
. . then $\text{arc}(AB) = \text{arc}(AC)$ .
(equal chords subtend equal arcs).

Therefore: . $\angle APB = \angle APC \quad\Rightarrow\quad AP \text{ bisects }\angle BPC.$

• Nov 10th 2008, 05:56 PM
bearej50
thank you very much!