Three Circles with Two Common Tangents

Let be a triangle circumscribed by the circle . Let be a point on ; form the line . Consider three other circles , , and with the common tangent , with inscribed in the triangle , and and tangent to both the segment and . Prove that , , and have two common tangents.


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


The statement holds for arbitrary points , , on . Moreover, the statement holds for an arbitrary point on . You can drag the vertices A, B and C and change the position of using a slider. This is problem 4 from the eleventh International Mathematical Olympiad (IMO) held in Bucharest, Romania, July 5–20, 1969.
[1] D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads, 1959–2009, 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.