Squares on a Line Segment

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 .


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+