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ortocenters of BCK,ACN coincide iff BN/AK =/tan^2 A, right ABC

Source: 2016 XIX All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p1

May 6, 2021

geometryorthocenterright triangleUkrainian TYM

Problem Statement

The points

K

and

N

lie on the hypotenuse

AB

of a right triangle

ABC

. Prove that orthocenters the triangles

BCK

and

ACN

coincide if and only if

\frac{BN}{AK}=\tan^2 A.