10182

# 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 .

### DETAILS

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

### PERMANENT CITATION

 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 » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.