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Thread: Proving?

  1. #1
    Member Kaloda's Avatar
    Sep 2010


    In the figure, Segment AB is a diameter of the smaller of the two concentric circles. Segments AP and BQ are tangent to the smaller circle at A and B, respectively.

    PROVE that Segments AB and PQ intersect at the center of the circle.
    Attached Thumbnails Attached Thumbnails Proving?-prob.png  
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  2. #2
    MHF Contributor
    Oct 2009
    For clarity, I made the circles very non-concentric in the following picture.

    Let O be the center of the circles, by definition. Then AP = BQ by the Pythagoras theorem. Therefore, $\displaystyle \triangle AOP= \triangle BOQ$. So, $\displaystyle \angle AOP=\angle BOQ$, which means that POQ is a straight line.
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  3. #3
    Super Member
    Nov 2007
    Trumbull Ct
    I did this proof as follows

    given two concentric circles,AB diameter of small circle, AP and BQ tangents to small circle @A and B

    1 extend PA and BQ to C and D points on the large circle
    2 TAngents @ A and B meet diameter AB @ 90 deg property of tangents
    3 PAC and QBD are parallel two lines perpendicular to a third line are parallel
    4 PAC and QBD are equal in lenght Each is a chord of large circle perpendicular to AB and equidistant to center of circles m the midpoint of AB
    5 PCQD is a rectangle CQ=PD =AB perpendicular distance between parallel are equal
    6 QP and CD are diagonals of the rectangle meeting AB @ m the middle of rectangle Property of rect diagonals
    7 QM =MP CM=DM AM- MB Mis the midpoint of AB and the point bisecting the three lines QP, CD, AB

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