here I shall collect in one place links solutions to old plane geometry problems from International Mathematical Olympiads, posted in this page
1959 IMO Problem 4 (HUN) (here - mine)
Construct a right triangle with given hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.
An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB; with the segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q; intersect at M and also at another point N: Let N' denote the point of intersection of the straight lines AF and BC:
(a) Prove that the points N and N' coincide.
(b) Prove that the straight lines MN pass through a fixed point S independent of the choice of M.
(c) Find the locus of the midpoints of the segments PQ as M varies between A and B.
1959 IMO Problem 4 (HUN) (here - mine)
Construct a right triangle with given hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.
proposed by Hungary
1959 IMO Problem 5 (ROM) (here - mine & Theodosis Giannopoulos)An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB; with the segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q; intersect at M and also at another point N: Let N' denote the point of intersection of the straight lines AF and BC:
(a) Prove that the points N and N' coincide.
(b) Prove that the straight lines MN pass through a fixed point S independent of the choice of M.
(c) Find the locus of the midpoints of the segments PQ as M varies between A and B.
proposed by Cezar Cosnita, Romania
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