# A question about triangles

Is a triangle uniquely determined by its area, perimeter, and sum of the reciprocals of its angles?

This is an open question, though it is supported by numerical evidence. This question is related to the inverse spectral problem of the Laplace operator in spectral geometry. In particular, the area, perimeter, and sums of the reciprocals of the angles are the first three eigenvalues of the Laplace operator. Note that a triangle, up to congruence, is determined by three pieces of information (for instance, the SSS theorem from high school geometry), which is why it makes sense to require at least three eigenvalues. We remark that all of the eigenvalues of the Laplace operator uniquely determine a triangle

On the other hand, right triangles are determined by two pieces of information (say the two legs). Thus one might guess that a right triangle is determined by the first two eigenvalues of the Laplace operator. Indeed, this is the case as we outline below that the area and perimeter of a triangle determine a right triangle. ${\bf Proposition.}$ Suppose $T$ is a right triangle with area $A$ and perimeter $P$. Then the two legs of the triangle are the solutions of the quadratic polynomial $-2P x^2 + (P^2 + 4A) x - 4AP = 0.$ ${\it Proof.}$ Let $x,y,z$ be the side lengths of $T$, where $z$ is the hypotenuse. Then $xy = 2A , x + y + z = P , x^2 + y^2 = z^2.$ The proposition follows from a modest computation. $\spadesuit$

For fun, one can check the solutions quadratic polynomial in the above proposition are invariant under the map $x \mapsto 2A / x$.