MathDB
Tangent semicircles and concyclic points

Source: India TST 2018, D1 P2

July 18, 2018
geometry

Problem Statement

Let A,B,CA,B,C be three points in that order on a line ℓ\ell in the plane, and suppose AB>BCAB>BC. Draw semicircles Γ1\Gamma_1 and Γ2\Gamma_2 respectively with ABAB and BCBC as diameters, both on the same side of ℓ\ell. Let the common tangent to Γ1\Gamma_1 and Γ2\Gamma_2 touch them respectively at PP and QQ, P≠QP\ne Q. Let DD and EE be points on the segment PQPQ such that the semicircle Γ3\Gamma_3 with DEDE as diameter touches Γ2\Gamma_2 in SS and Γ1\Gamma_1 in TT. [*]Prove that A,C,S,TA,C,S,T are concyclic. [*]Prove that A,C,D,EA,C,D,E are concyclic.
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