2000 JBMO Problems/Problem 3
A half-circle of diameter is placed on the side of a triangle and it is tangent to the sides and in the points and respectively. Prove that the intersection point between the lines and lies on the altitude from of the triangle .
We begin by showing that is the circumcenter of :
Consider the configuration where is obtuse and diameter lies on extended line .
Let be the midpoint of diameter . Let meet at .
Let us define and
By applying Tangent Chord Angle theorem, we get: and
Now, , and since is a cyclic quadrilateral, we have
Now , so
Similarly, we have , so
Thus, we have
Also, (Since and are tangents to the same circle)
From the above 2 results, it readily follows that is the circumcenter of .
Thus, we have , and so
So is perpendicular to , hence lies on the altitude from of the triangle .