Three Circles with Two Common Tangents

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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.

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



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.

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