Circle Proofs

[Summer12]

Could someone please help me with this circle proof? I have test on this soon but I'm not exactly sure how to do circle proof or proofs in general.... Thanks in advance & any help is apperciated.

 

[wjm11]

Here are a couple of hints:

Draw the radius from circle center to point P. This radius and the tangent line are perpendicular to each other.

Draw the perpendicular bisector of segment XY. It will/must pass through the center of the circle.

Since XY and the tangent line are parallel, these two segments are co-linear.

Hope that helps.

 

[soroban]

Hello, Summer12!

\(\displaystyle \text{Given: }\:AP\!B\text{ is tangent to circle }O\text{ at }P,\;X\!Y \parallel AP\!B\)

\(\displaystyle \text{Prove that }\Delta P\!XY\text{ is isosceles.}\)

              P
    A - - - * o * - - - B
        *    / \    *
      *     /   \     *
     *     /     \     *
          /       \
    *    /         \    *
    *   /           \   *
    *  /             \  *
      /               \
   X o- - - - - - - - -o Y
      *               *
        *           *
            * * *

\(\displaystyle \begin{array}{ccccccc}1. & AB \parallel XY && 1. & \text{Given} \\ \\[-3mm] 2. & \text{arc(PX)} = \text{arc(PY)} && 2. & \text{Parallel lines intercept equal arcs} \\ \\[-3mm] 3. & P\!X \,=\,PY && 3. & \text{Equal arcs} = \text{equal chords} \\ \\[-3mm] 4. & \Delta P\!XY\text{ is isosceles} && 4. & \text{D{e}f. isosceles triangle} \end{array}\)

 

[wjm11]

I like Soroban's way better!