2007 USAMO Problems/Problem 6
Contents
Problem
(Kiran Kedlaya, Sungyoon Kim) Let be an acute triangle with , , and being its incircle, circumcircle, and circumradius, respectively. Circle is tangent internally to at and tangent externally to . Circle is tangent internally to at and tangent internally to . Let and denote the centers of and , respectively. Define points , , , analogously. Prove that with equality if and only if triangle is equilateral.
Solutions
Solution 1
Lemma.
Proof. Note and lie on since for a pair of tangent circles, the point of tangency and the two centers are collinear.
Let touch , , and at , , and , respectively. Note . Consider an inversion, , centered at , passing through , . Since , is orthogonal to the inversion circle, so . Consider . Note that passes through and is tangent to , hence is a line that is tangent to . Furthermore, because is symmetric about , so the inversion preserves that reflective symmetry. Since it is a line that is symmetric about , it must be perpendicular to . Likewise, is the other line tangent to and perpendicular to .
Let and (second intersection).
Let and (second intersection).
Evidently, and . We want: by inversion. Note that , and they are tangent to , so the distance between those lines is . Drop a perpendicular from to , touching at . Then . Then . So . Note that . Applying the double angle formulas and , we get
End Lemma
The problem becomes: which is true because , equality is when the circumcenter and incenter coincide. As before, , so, by symmetry, . Hence the inequality is true iff is equilateral.
Comment: It is much easier to determine by considering . We have , , , and . However, the inversion is always nice to use. This also gives an easy construction for because the tangency point is collinear with the intersection of and .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See also
- <url>viewtopic.php?t=145852 Discussion on AoPS/MathLinks</url>
2007 USAMO (Problems • Resources) | ||
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