Problem of the Week

Problem E

Terry’s Triangles

Terry is drawing isosceles triangles with side lengths $$a$$, $$b$$, and $$c$$ such that \begin{aligned} a&=y-x\\ b&=x+z\\ c&=y-z\end{aligned} where $$x$$, $$y$$, and $$z$$ are positive integers and $$x+y+z<10$$.
Find all the possible triples $$(a,b,c)$$ that satisfy this.