2006 Canadian MO Problems/Problem 5
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. ! Double superscript.) where the tangent at intersects the extended lines and . Similarly for on arc and on arc . Prove that triangle is equilateral.
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 prove that triangle is equilateral. To see this, note that Hence as well, so triangle is equilateral as desired.
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