MathDB
x,y,z collinear

Source: Iran TST 2015,second exam,second day,problem 4

June 4, 2015
geometry

Problem Statement

Let △ABC\triangle ABC be an acute triangle. Point ZZ is on AA altitude and points XX and YY are on the BB and CC altitudes out of the triangle respectively, such that: ∠AYB=∠BZC=∠CXA=90\angle AYB=\angle BZC=\angle CXA=90 Prove that XX,YY and ZZ are collinear, if and only if the length of the tangent drawn from AA to the nine point circle of △ABC\triangle ABC is equal with the sum of the lengths of the tangents drawn from BB and CC to the nine point circle of △ABC\triangle ABC.
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