2001 JBMO Problems/Problem 2
Let be a triangle with and . Let be an altitude and be an interior angle bisector. Show that for on the line we have . Also show that for on the line we have .
Assume that . Since is an angle bisector of Thus, by AAS Congruency, which results in But so by proof by contradiction,
Assume that . Draw points and on and respectively such that is a rectangle. That means and are cyclic quadrilaterals, which means that
Because is bisected by , by the Angle Bisector Theorem, we have so by SAS Congruency, we have making and a square. This also means that so by AAS Congruency, making However, it is given that so by proof by contradiction.
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