MathDB
cyclic quadrilaterals and angle bisector wanted, tangent circles

Source: 2008 Indonesia TST stage 2 test 2 p3

December 14, 2020
geometrycyclic quadrilateralangle bisectortangent circles

Problem Statement

Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles that tangents each other at point NN, with Γ2\Gamma_2 located inside Γ1\Gamma_1. Let A,B,CA, B, C be distinct points on Γ1\Gamma_1 such that ABAB and ACAC tangents Γ2\Gamma_2 at DD and EE, respectively. Line NDND cuts Γ1\Gamma_1 again at KK, and line CKCK intersects line DEDE at II. (i) Prove that CKCK is the angle bisector of ∠ACB\angle ACB. (ii) Prove that IECNIECN and IBDNIBDN are cyclic quadrilaterals.
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