I'm trying to answer this question but I'm fully stuck. I've tried to draw the diagram but it still is a mystery to me. Here is the problem:
a) Prove that the $2$-$3$-$4$ triangle has the property that one of the triangles formed by the lines containing two of its sides and the angle bisector of one of its external angles is similar to the original triangle.
b) Find all triangles with no side length greater than $10$ with this property.
c) Prove that there exists a triangle with this property with longest side length $n$ iff $n$ is not divisible by the square of any prime.