Math CentralQuandaries & Queries


Question from Kimberly, a student:

I'm trying to write a proof for the following: If all altitudes are equal in an equilateral triangle then all sides are equal.

Please help!

Hi Kimberly.

Why don't you write the proof based on the area of a triangle?

The area of a triangle is half the base times the altitude.

The area is a constant and the altitudes are the same in all three circumstances, so the bases must be the same as well.

You have to write it up formally, of course, but that's a quick way to prove it.

Stephen La Rocque.

Kimberley wrote back

Thank you for that response. However, I also need to know how to prove that all sides are congruent given the fact that all altitudes are congruent.
I would really appreciate it if you could help me with this problem also.

Thank you, Kim


Suppose that the triangle is ABC.

Write the expression for the area of the triangle using AB as the base.
Write the expression for the area of the triangle using BC as the base.

Since the area is the same regardless which side you choose as the base, the two expressions you wrote are equal. What does this tell you about AB and BC?


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