MathDB

3

Part of 2017 Taiwan TST Round 3

Problems(1)

NPC lies on the A-Napoleon line of another triangle

Source: 2017 Taiwan TST Round 3, Test 1, Problem 3

4/26/2017
Given a △ABC \triangle ABC with circumcircle Γ. \Gamma. Let A′ A' be the antipode of A A in Γ \Gamma and D D be the point s.t. △BCD \triangle BCD is an equilateral triangle (A A and D D are on the opposite side of BC BC ). Let the perpendicular from A′ A' to A′D A'D cuts CA, CA, AB AB at E, E, F, F, resp. and T T be the point s.t. △ETF \triangle ETF is an isosceles triangle with base EF EF and base angle 30∘ 30^{\circ} (A A and T T are on the opposite side of EF EF ). Prove that AT AT passes through the 9-point center of △ABC. \triangle ABC.
Proposed by Telv Cohl
geometrygeometry proposed