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Circumcircle and exterior line

Source: Czech and Slovak Olympiad 1967, National Round, Problem 4

July 1, 2024
geometrycircumcircle

Problem Statement

Let ABCABC be an acute triangle, kk its circumcirle and mm a line such that m∩k=∅,m∥BC.m\cap k=\emptyset, m\parallel BC. Denote DD the intersection of mm and ray AB.AB. a) Let XX be an inner point of the arc BCBC not containing AA and denote YY the intersection of lines m,CX.m,CX. Show that A,D,X,YA,D,X,Y are concyclic and name this circle κ\kappa. b) Determine relative position of κ\kappa and mm in case when C,D,XC,D,X are collinear.
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