MathDB
fixed circle

Source: Problem 6 VMO 2013 day 2

January 12, 2013
geometrycircumcirclegeometry proposed

Problem Statement

Let ABCABC be a cute triangle.(O)(O) is circumcircle of △ABC\triangle ABC.DD is on arc BCBC not containing AA.Line △\triangle moved through HH(HH is orthocenter of △ABC\triangle ABC cuts circumcircle of △ABH\triangle ABH,circumcircle △ACH\triangle ACH again at M,NM,N respectively. a.Find △\triangle satisfy SAMNS_{AMN} max b.d1,d2d_{1},d_{2} are the line through MM perpendicular to DBDB,the line through NN perpendicular to DCDC respectively. d1d_{1} cuts d2d_{2} at PP.Prove that PP move on a fixed circle.
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