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Triangles have the same perimeter from the two S circles

Source: CentroAmerican 2003

November 29, 2010
geometryperimeterparallelogramgeometry proposed

Problem Statement

S1S_1 and S2S_2 are two circles that intersect at two different points PP and QQ. Let ℓ1\ell_1 and ℓ2\ell_2 be two parallel lines such that ℓ1\ell_1 passes through the point PP and intersects S1,S2S_1,S_2 at A1,A2A_1,A_2 respectively (both distinct from PP), and ℓ2\ell_2 passes through the point QQ and intersects S1,S2S_1,S_2 at B1,B2B_1,B_2 respectively (both distinct from QQ). Show that the triangles A1QA2A_1QA_2 and B1PB2B_1PB_2 have the same perimeter.
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