MathDB
Perpendicular Line Geometry

Source: KJMO 2005

May 20, 2018
geometrycircumcircle

Problem Statement

In △ABC\triangle ABC, let the bisector of ∠BAC\angle BAC hit the circumcircle at MM. Let PP be the intersection of CMCM and ABAB. Denote by (V,WX,YZ)(V,WX,YZ) the intersection of the line passing VV perpendicular to WXWX with the line YZYZ. Prove that the points (P,AM,AC),(P,AC,AM),(P,BC,MB)(P,AM,AC), (P,AC,AM), (P,BC,MB) are collinear.
In isosceles triangle APXAPX with AP=AXAP=AX, select a point MM on the altitude. PMPM intersects AXAX at CC. The circumcircle of ACMACM intersects APAP at BB. A line passing through PP perpendicular to BCBC intersects MBMB at ZZ. Show that XZXZ is perpendicular to APAP.
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