Question

Prove a triangle having two equal altitudes is isosceles

Answer 1

Proof 
Let ABC be a triangle with altitudes AD and BE of equal length                         
We need to prove that the sides AC and BC are of equal length. 

Consider the triangles ADC and BEC.
They are the right triangles with the common angle ACB.
The angles CAD and CBE are congruent as the complementary angles to the angle ACB.
Thus, the triangles ADC and BEC have congruent sides AD and BE,
as well as congruent angles ADC and BEC (right angles) and congruent 
angles CAD and CBE.
Therefore, the triangles ADC and BEC are congruent, in accordance 
to the postulate P2 (ASA) (see the lesson Congruence tests for triangles of the 
topic Triangles in the section Geometry in this site).
Hence, the straight segments AC and BC are of equal length as the corresponding 
sides of these triangles. 
The proof is completed