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A spiral configuration

Source: INMO 2020 P1

January 19, 2020
Spiral Similaritycirclesgeometrydumpty point

Problem Statement

Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles of unequal radii, with centres O1O_1 and O2O_2 respectively, intersecting in two distinct points AA and BB. Assume that the centre of each circle is outside the other circle. The tangent to Γ1\Gamma_1 at BB intersects Γ2\Gamma_2 again in CC, different from BB; the tangent to Γ2\Gamma_2 at BB intersects Γ1\Gamma_1 again at DD, different from BB. The bisectors of ∠DAB\angle DAB and ∠CAB\angle CAB meet Γ1\Gamma_1 and Γ2\Gamma_2 again in XX and YY, respectively. Let PP and QQ be the circumcentres of triangles ACDACD and XAYXAY, respectively. Prove that PQPQ is the perpendicular bisector of the line segment O1O2O_1O_2.
Proposed by Prithwijit De
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