Question

If two altitudes of the triangle are equal prove that it is an isoceles triangle.​

Answer 1

Explanation:

Let ABC be a triangle with altitudes AD and BE of equal length as shown in figure.

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 angle) and congruent angles CAD and CBE.

Therefore, the triangles ADC and BEC are congruent, in accordance to the postulate P2 (ASA). 

Thus, the straight segments AC and BC are of equal length as the corresponding sides of these triangles and in other words, the triangle is isosceles.

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