Two Circles in a Quadrilateral

Geometry Level 3

Two circles intersect at points M M and N N , and quadrilateral A B C D ABCD is constructed so that A A , M M , N N , and C C are collinear, point P P on A B AB and point Q Q on B C BC are points of tangency to one of the circles, and point R R on C D CD and point S S on A D AD are points of tangency to the other circle.

If A B = 12 AB = 12 , C D = 10 CD = 10 , and A D = 8 AD = 8 , find B C BC .


The answer is 14.

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1 solution

Mr. India
Mar 20, 2019

Tangents from same point are equal in length.

A S 2 = A M . A N = A P 2 AS^2=AM.AN=AP^2 So, A S = A P AS=AP ( why? )

Let A S = A P = x AS=AP=x

So, S D = 8 x , P B = 12 x SD=8-x, PB=12-x

Now, S D SD and R D RD are tangents from D D

So, S D = R D = 8 x SD=RD=8-x

So, C R = 10 ( 8 x ) = 2 + x CR=10-(8-x)=2+x ( C R = C D R D ) (CR=CD-RD)

Now, C R = C Q = 2 + x CR=CQ=2+x

Also, P B = Q B = 12 x PB=QB=12-x

So, B C = B Q + C Q BC=BQ+CQ

= 12 x + 2 + x = 14 =12-x+2+x=\boxed{\boxed{14}}

AS and AP are equal (and CR and CQ are equal), but not for the reason you gave. The theorem you used (that tangents from the same point are equal in length) refers to tangents of the same circle. In this problem, AS and AP (and CR and CQ) are tangents to different circles.

David Vreken - 2 years, 2 months ago

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Thank you for pointing out! I have edited the answer. (wow, someone even upvoted for it before edit!)

Mr. India - 2 years, 2 months ago

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Yes, that looks better!

David Vreken - 2 years, 2 months ago

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