Question

Are all isosceles triangles equilateral?

Answer 1

No.
Isosceles triangles are those that have two sides to be of equal length, while equilateral triangles are those that have all three sides of equal length.

Explanation:

Theorem
There exist isosceles triangles that are not equilateral.

Proof
A triangle with a base of 10 cm and two other sides each of 20 cm is isosceles, but not equilateral.

This counter example proves that not all isosceles triangles are equilateral.