2006 Canadian MO Problems/Problem 5
Problem
The vertices of a right triangle inscribed in a circle divide the circumference into three arcs. The right angle is at , so that the opposite arc is a semicircle while arc and arc are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines and . More precisely, the point on arc is the midpoint of the segment joining the points and $D^\prime^\prime$ (Error compiling LaTeX. Unknown error_msg) where the tangent at intersects the extended lines and . Similarly for on arc and on arc . Prove that triangle is equilateral.
Solution
Let the intersection of the tangents at and , and , and be labeled , respectively. It is a well-known fact that in a right triangle with the midpoint of hypotenuse , triangles and are isosceles. Now, we do some angle-chasing: whence we conclude that Next, we will prove that triangle is equilateral. To see this, note that Then, as well, and we are done.
See also
2006 Canadian MO (Problems) | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 | Followed by Last question |