2006 Romanian NMO Problems/Grade 7/Problem 3
In the acute-angle triangle we have . The points and are the feet of the altitudes from and , and is the orthocenter of the triangle. We consider the points and on the segments and such that . Prove that
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a) Note that quadrilateral is cyclic, because . Thus and . Similarly . Therefore and . However is a triangle, so and . By Pythagorean theorem, . However , so , and thus , or .
b) , , .