Is there an isosceles triangle with integers sides?

Question

Is there an isosceles triangle with integers sides?

I saw this question in a section of integers as sum of squares, on a number theory book. Could someone help me?

Answer 1

Consider any integral-sided equilateral triangle.

Equilateral $\implies$ Isosceles.

Answer 2

An isosceles triangle with sides $a,a,b$ ($>0$) exists if and only if $2a>b$ (i.e. satisfies Triangle Inequality). If it exists, by the Law of Cosines the angle between the equal sides is equal to $\arccos\left(1-\frac{b^2}{2a^2}\right)$.

So yes, an isosceles triangle with positive integer sides exists. If positive integers $m,n$ satisfy $2m>n$, then an isosceles triangle with sides $m,m,n$ exists.