Squares on a Line Segment

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A segment is given and on it the point . On the same side of , the squares and are constructed. The circumcircles of the two squares, whose centers are and , intersect at and another point .


(a) Prove that lines and intersect at .

(b) Prove that all such constructed lines pass through the same point , regardless of the selection of .

(c) Find the locus of the midpoints of all segments , as varies along the segment .


Contributed by: Jaime Rangel-Mondragon (July 2013)
Open content licensed under CC BY-NC-SA



This is the first of a series of Demonstrations dedicated to showcasing a sample of problems posed for the International Mathematical Olympiads (IMO), the most important and prestigious annual mathematical competition for high school students, which began in 1959. (In 1980, financial problems caused no country to volunteer to host it.)

The problems chosen have an intrinsic geometrical appeal and provide an interesting programming challenge, met with the framework provided by Mathematica. The statement of the problems we present follows the original ones, in which the proof of a series of assertions is required.

Our goal is to aid in the visual understanding of the problem and of its assertions. Sometimes a visual hint of the proof itself is provided; for instance, the dotted circle having as its diameter passes through and , and the locus of part (c) is indicated in purple. This problem was taken from the first IMO in Bucharest-Brasov, Romania, July 23-31, 1959, problem 5 [1].


[1] D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium, 2nd ed., New York: Springer, 2011.

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