A triangle with an angle, the altitude through that angle, and perimeter given. Is the triangle unique? (from Polya's "How to Solve It")

Question

I was reading Polya's "How to Solve It" when I came across the following problem.

Construct a triangle with an angle, the length of altitude through that angle, and the perimeter of the triangle given.

I wasn't able to prove that such a triangle would be unique. Is the given data enough to prove the uniqueness of the triangle?

Secondly how do we construct such a triangle? Rather I never understood the importance of Euclidean constructions in mathematics using only a straight ruler and a compass. What is the need of such constructions?

Thanks in advance.

Answer 1

The idea is that for a given angle, say $\angle BAC$, the locus of the foot of an altitude from $A$ that has a fixed altitude length $h$ is an arc of a circle of radius $h$. The perpendicular to this radius will intersect $AB$ and $AC$ at $B$ and $C$, respectively. It should not be too difficult to derive an expression for the perimeter of $\triangle ABC$ for a given $h$ and $\angle BAC$ as a function of some angle $\theta$ that the altitude makes with the bisector of $\angle BAC$. Intuitively, the triangle of least perimeter will occur when $\theta = 0$, so up to reflection, such a triangle is unique (or nonexistent if the desired perimeter is too small).