1989 USAMO Problems/Problem 4
Let be an acute-angled triangle whose side lengths satisfy the inequalities . If point is the center of the inscribed circle of triangle and point is the center of the circumscribed circle, prove that line intersects segments and .
Consider the lines that pass through the circumcenter . Extend , , to ,, on ,,, respectively.
We notice that passes through sides and if and only if belongs to either regions or .
Since , we let , , .
Since divides angle into two equal parts, it must be in the region marked by the of angle , so is in .
Similarly, is in and . Thus, is in their intersection, . From above, we have passes through and .
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