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Math Help - 3 Circle Geometry Proofs

  1. #1
    Member
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    3 Circle Geometry Proofs

    Hi, here are 3 sums which I need help in proving.

    1) PQ is a tangent to the circle at A and O is the center of the circle. If BC is parallel to PQ, prove that angle BOA = 2x angle CAQ.


    2) AC is a tangent at A to the circle, center O. BC is perpendicular to CA. Prove that BC is parallel to OA and AB bisects angle OBC.


    3) The chords AB and CD intersect at E and the tangent PA is parallel to chord CD. Prove that AB bisects angle CBD.


    Thanks for your explanations.
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  2. #2
    MHF Contributor
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    Hello

    1) AO is an axis of symmetry of the figure => angle BOA=angle COA
    OC=OA => triangle COA is isosceles => angle OCA=angle OAC

    Sum of angles of triangle COA=180
    angle COA + angle OCA + angle OAC = 180
    angle COA + 2 x angle OAC = 180 (1)

    PQ is tangent to the circle => angle OAQ = 90
    But angle OAQ = angle OAC + angle CAQ = 90
    Therefore angle OAC = 90 - angle CAQ (2)

    (1) and (2)
    angle COA + 2 x (90 - angle CAQ) = 180
    angle COA - 2 x angle CAQ = 0
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