If we have a triangle where the Perimeter >0 and the Area >0 , and Area=Perimeter, what special condition must the angles of this triangle satisfy for this to happen?
I've done a bit of research and have found information on Heron's forumla and on integral triangles but I'm at a loss at how to figure this problem out.
Through research I have found that if the lengths of the sides of the triangle are either {6,8,10),(5,12,13),(9,10,17),(7,15,20), or(6,25,29) we get a triangle that fits the requirements (im not sure if this information is relevant) but the question is about the "special condition" that the angles must meet to create such a triangle not the sides.
I'm looking for both an answer and an explanation, any help is greatly appreciated. Please let me know if I left anything out (that's all the information I was given) or if something is unclear.
Hint: If you scale the triangle by a factor of $\alpha$, the perimeter is scaled by $\alpha$ and the area is scaled by $\alpha^2$.
Hence, given any triangle, there is a unique triangle that is similar to it, which satisfies perimeter = area.