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Thread: Circle Theorems

  1. #1
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    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.
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  2. #2
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    Hello, bearej50!

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

    Prove: If $\displaystyle P$ is any point on minor arc $\displaystyle BC$, then ray $\displaystyle PA$ bisects $\displaystyle \angle BPC.$
    Code:
                    A
                  * o *
              *    / \    *
            *     /   \     *
           *     /   o \     *
                /       \
          *    /         \    *
          *   /       o   \   *
          *  /             \  *
            /               \
          B*- - - - - -o- - -*C
            *               *
              *           *
                  * * * o
                         P

    Draw chords $\displaystyle PB$ and $\displaystyle PC$.

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

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

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

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  3. #3
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    thank you very much!
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