A triangle with an angle, the altitude through that angle, and perimeter given. Is the triangle unique? (from Polya's "How to Solve It") 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.
 A: There is a standard method to reduce questions of this type to ones about uniqueness and existence of solutions for systems of equations. In your case, if the triangle is $ABC$ and we are given the angle $\gamma$, altitude length $k$ there and perimeter $2s$, then the equations (for $p,q,c$) are:
$q=k$, $qc = \sin \gamma \sqrt{(p^2+q^2)((p-c)^2+q^2)}$ and $2s=\sqrt{p^2+q^2}+\sqrt{(p-c)^2+q^2}$.  You could try to work this out by hand, or use Mathematica if available. This should give you uniqueness and the precise relation between the given quantities required to ensure uniqueness.
Edit: I forgot to mention that then $A=(0,0)$, $B=(c,0)$ and $C=(p,q)$.
The requirement of ruler and compass is, of course, no longer of practical relevance. However, it would have been, for obvious reasons, in antiquity.  You will have noticed that most technical and architectural  plans until recently mainly used consructible curves (lines, circles, occasionally ellipses and parabolas) in contrast to post-modern architecture where, due to the use of software, pretty well anything goes.
A: 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).
