Circle Theorems

bearej50 · #1 $Old$ 11-10-2008, 11:41 AM

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.

Soroban · #2 $Old$ 11-10-2008, 01:58 PM

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

bearej50 · #3 $Old$ 11-10-2008, 06:56 PM

thank you very much!