# Thread: Circle Theorems

1. ## 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.

2. Hello, bearej50!

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

3. thank you very much!