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Sum of radii of incircle and excircle is the length of [FA]

Source: German DEMO 2008, grade 11-13, round 4, problem 2

May 22, 2008
geometryrectanglegeometry proposed

Problem Statement

The triangle △SFA \triangle SFA has a right angle at F F. The points P P and Q Q lie on the line SF SF such that the point P P lies between S S and F F and the point F F is the midpoint of the segment [PQ] [PQ]. The circle k1 k_1 is th incircle of the triangle △SPA \triangle SPA. The circle k2 k_2 lies outside the triangle △SQA \triangle SQA and touches the segment [QA] [QA] and the lines SQ SQ and SA SA. Prove that the sum of the radii of the circles k1 k_1 and k2 k_2 equals the length of [FA] [FA].
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