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Quadrilateral admits an excircle

Source: India IMOTC Practice Test 2 Problem 3

May 31, 2024
geometry

Problem Statement

Let ABCDABCD be a convex quadrilateral which admits an incircle. Let ABAB produced beyond BB meet DCDC produced towards CC, at EE. Let BCBC produced beyond CC meet ADAD produced towards DD, at FF. Let GG be the point on line ABAB so that FG∥CDFG \parallel CD, and let HH be the point on line BCBC so that EH∥ADEH \parallel AD. Prove that the (concave) quadrilateral EGFHEGFH admits an excircle tangent to EG‾,EH‾,FG→,FH→\overline{EG}, \overline{EH}, \overrightarrow{FG}, \overrightarrow{FH}.
Proposed by Rijul Saini
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