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.

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

thank you very much!