MathDB
2 intersecting circles, tangents, intersections, equal segments, tangent wanted

Source: Danube 2014 Seniors P1

September 8, 2018
geometrycirclesTangents

Problem Statement

Two circles γ1\gamma_1 and γ2\gamma_2 cross one another at two points; let AA be one of these points. The tangent to γ1\gamma_1 at AA meets again γ2\gamma_2 at BB, the tangent to γ2\gamma_2 at AA meets again γ1\gamma_1 at CC, and the line BCBC meets again γ1\gamma_1 and γ2\gamma_2 at D1D_1 and D2D_2, respectively. Let E1E_1 and E2E_2 be interior points of the segments AD1AD_1 and AD2AD_2, respectively, such that AE1=AE2AE_1 = AE_2. The lines BE1BE_1 and ACAC meet at MM, the lines CE2CE_2 and ABAB meet at NN, and the lines MNMN and BCBC meet at PP. Show that the line PAPA is tangent to the circle ABCABC.
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