perplexus.info :: Geometry : Collinear intersections

Collinear intersections (Posted on 2009-07-13)

Two smaller circles of radii a and b are internally tangent to a larger circle, of radius r, at points P and Q respectively. The smaller circles intersect at points S and T.

If P, S, and Q are collinear, prove that r = a+b.

Submitted by Bractals

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Solution: (Hide)

Let A, B, and R be the centers of the circles with radii a, b, and r respectively. Conctruct line segments PAR, QBR, PSQ, AS, and BS. Note that PAS, PRQ, and SBQ are similar isosceles triangles. /PAS = /PRQ ==> AS || RBQ (1) /SBQ = /PRQ ==> BS || RAP (2) (1) & (2) ==> ARBS is a parallelogram ==> |AR| = |SB| ==> r - a = b ==> r = a+b

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): proof	Charlie	2009-07-14 02:34:14
re: proof	Bractals	2009-07-14 01:16:25
proof	Charlie	2009-07-13 17:53:05

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