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Math Help - Geometry proof: Two tangents to a circle

  1. #1
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    Geometry proof: Two tangents to a circle

    I found this little proof in a book I was practicing out of. Does anyone have a proof for this? It must be so simple! Gaahh!

    Let y be a circle, l and m be two nonparallel lines that are tangent to y at the points P and Q, and let A be the point of intersection of l and m. Prove that:
    i) O lies on the bisector of angle(PAQ)
    ii) PA = QA
    iii) The line PQ is perpendicular to the line OA

    I know Triangle POQ is isosceles, and OP = PQ, etc etc all the things about that triangle, but I don't know how to work the point A into the mix, nor do I know anything (I don't think) about Triangle PQA. Any help here?
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  2. #2
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    You need to prove that APQO is a kite-shape quadrangle, you can do so by noticing that the angle between the radius and the tangent is 90 degrees, the triangle PQO is isosceles, so the angles OQP and OPQ are equal and thus AQP and APQ are equal as well, can you finish from here?
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  3. #3
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    Quote Originally Posted by paupsers View Post
    I found this little proof in a book I was practicing out of. Does anyone have a proof for this? It must be so simple! Gaahh!

    Let y be a circle, l and m be two nonparallel lines that are tangent to y at the points P and Q, and let A be the point of intersection of l and m. Prove that:
    i) O lies on the bisector of angle(PAQ)
    ii) PA = QA
    iii) The line PQ is perpendicular to the line OA

    I know Triangle POQ is isosceles, and OP = PQ, etc etc all the things about that triangle, but I don't know how to work the point A into the mix, nor do I know anything (I don't think) about Triangle PQA. Any help here?
    Of course, a tangent to a circle is perpendicular to the radius at the point of tangency. Triangles OPA and OQA are right triangles. Further, the fact that POQ is, as you say, isosceles shows that PQ is perpendicular to OA.
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