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Italian Mathematical Olympiad 2008

Source: Problem 5

August 23, 2008
geometry proposedgeometry

Problem Statement

Let ABC ABC be a triangle, all of whose angles are greater than 45∘ 45^{\circ} and smaller than 90∘ 90^{\circ}. (a) Prove that one can fit three squares inside ABC ABC in such a way that: (i) the three squares are equal (ii) the three squares have common vertex K K inside the triangle (iii) any two squares have no common point but K K (iv) each square has two opposite vertices onthe boundary of ABC ABC, while all the other points of the square are inside ABC ABC. (b) Let P P be the center of the square which has AB AB as a side and is outside ABC ABC. Let rC r_{C} be the line symmetric to CK CK with respect to the bisector of ∠BCA \angle BCA. Prove that P P lies on rC r_{C}.
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