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Geometry Mathley 3.2 fixed point

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June 7, 2020
fixedFixed pointequal ratiogeometry

Problem Statement

Given a triangle ABCABC, a line δ\delta and a constant kk, distinct from 00 and 1,M1,M a variable point on the line δ\delta. Points E,FE, F are on MB,MCMB,MC respectively such that ME‾MB‾=MF‾MC‾=k\frac{\overline{ME}}{\overline{MB}} = \frac{\overline{MF}}{\overline{MC}} = k. Points P,QP,Q are on AB,ACAB,AC such that PE,QFPE, QF are perpendicular to δ\delta. Prove that the line through MM perpendicular to PQPQ has a fixed point.
Nguyễn Minh Hà
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