Pythagorean Condition in An Isosceles Right Triangle: What is this about?
A Mathematical Droodle
What if applet does not run? 
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
The applet attempts to suggest the following problem:
Consider the right isosceles triangle ABC and the points M, N on the hypothenuse BC in the order B, M, N, C such that 
What if applet does not run? 
The configuration appears in a problem introduced in the National Olympiad 2001, in Romania, by Mircea Fianu.
Let's rotate ΔABC 90° counterclockwise around A. Since, by construction,
ΔNCM' being right, we have

Therefore, NM' = NM and, by SSS, ΔAMN = ΔAM'N. So that angles MAN and NAM' are equal and both are equal 45°.
(This problem admits a nice generalization.)
References
 W. G. Boskoff and B. D. Suceava, A Projectivity Characterized by the Pythagorean Relation, Forum Geometricorum, Volume 6 (2006) 187–190.
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
67246868