# Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.

Given: In ΔABC,

AB = AC = BC

To prove: 3 AB2 = 4 AD2

Theorem Used:

1) If corresponding side and two angles of two triangles are equal the triangles are said to have SAS congruency.

2) Pythagoras theorem:

In a right-angled triangle, the squares of the hypotenuse is equal to the sum of the squares of the other two sides.

Proof: AB = AC (given)

B = C (each 60°)

BD = DC As ΔADB is a right triangle right angled at D.

By Pythagoras theorem,

AB2 = AD2 + (1/2 BC)2

AB2 = AD2 + (1/4 BC2)

As BC = AB

AB2 = AD2 + (1/4 AB2)  Hence proved.

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