Question

Answer with proper explanation.
1.If all three altitudes of a triangle are equal then prove that it is an equilateral triangle

Answer 1

Dear friend,

An altitude of a triangle is the line drawn from any side forming a 90° angle.

In an equilateral triangle, all sides and all angles are equal to each other. This is why the altitude of each side which is equal is also equal to all the other altitude of the triangle.

Hope it helps !!!

Answer 2

let the traingle be ∆ABC and its altitudes to AB, BC, CA be D, E, F respectively.
therefore
CD = AE = BF = x units
as area of a triangle = ½×base×height
therefore ar(ABC) = ½×AB×CD
= ½×AB×x
similarity ar(ABC) = ½×AE×BC = ½×BC×x
and ar(ABC) = ½×BF×CA = ½×CA×x
as ar(ABC) = ar(ABC)
therefore
½xAB = ½xBC = ½xCA
so AB = BC = CA
as all sides are equal
therefore it's an equilateral triangle