View source for Mock AIME 3 Pre 2005 Problems/Problem 14

==Problem==
Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite sides of line <math>l</math>, and are both tangent to <math>l</math> at <math>P</math>. <math>\omega_3</math> passes through <math>P</math>, intersecting <math>l</math> again at <math>Q</math>. Let <math>A</math> and <math>B</math> be the intersections of <math>\omega_1</math> and <math>\omega_3</math>, and <math>\omega_2</math> and <math>\omega_3</math> respectively. <math>AP</math> and <math>BP</math> are extended past <math>P</math> and intersect <math>\omega_2</math> and <math>\omega_1</math> at <math>C</math> and <math>D</math> respectively. If <math>AD = 3, AP = 6, DP = 4,</math> and <math>PQ = 32</math>, then the area of triangle <math>PBC</math> can be expressed as <math>\frac{p\sqrt{q}}{r}</math>, where <math>p, q,</math> and <math>r</math> are positive integers such that <math>p</math> and <math>r</math> are coprime and <math>q</math> is not divisible by the square of any prime. Determine <math>p + q + r</math>.

==Solution==
Invert about a circle with radius 1 and center P. Note that since all relevant circles and lines go through P, they all are transformed into lines, and <math>\omega_1,\omega_2, l</math> are all tangent at infinity (i.e. parallel). That was the crux move; some more basic length chasing using similar triangles gets you the answer.
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==See Also==
{{Mock AIME box|year=Pre 2005|n=3|num-b=13|num-a=15}}