Problem
GEO-B1-M02-P019 Equal Altitudes
#19
★★★☆☆ Level 3 of 5
In triangle \(ABC\), altitudes \(BH\) and \(CK\), drawn to sides \(AC\) and \(AB\), are equal. Prove that \(AB=AC\).
Consider right triangles \(ABH\) and \(ACK\).
Triangles \(ABH\) and \(ACK\) are right triangles. They have \(BH=CK\) by the condition and \(\angle BAH=\angle CAK\), because both are equal to angle \(A\) of the original triangle. Therefore the third angles are also equal: \(\angle ABH=\angle ACK\). Sides \(BH\) and \(CK\) lie between equal angles, so by ASA triangles \(ABH\) and \(ACK\) are congruent. Therefore their hypotenuses are equal: \(AB=AC\).
This problem shows how altitudes become legs of right triangles.