Hyperbolic straight line segment of shortest length is perpendicular

Question

Given a hyperbolic straight line $C$ and a point $p$ not on $C$, the hyperbolic straight line segment of shortest length connecting $ p $ and $C$ is perpendicular to $C$.

I've tried considering the hyperbolic straight line segment connecting $-1$ and $1$ in the upper half plane, but I'm going nowhere.

Answer 1

We need to prove that the hypotenuse is the longest side of a right triangle.

This follows from these results in Euclid's Elements:

Euclid (I,17): In any triangle the sum of any two angles is less than two right angles.

Euclid (I,18): In any triangle, the angle opposite the greater side is greater.

The Euclidean proofs are still valid in hyperbolic geometry. The primary result that the proofs depend on (which is valid in hyperbolic geometry, but not spherical geometry) is:

Euclid (I,16): In any triangle, if one of the sides if produced, then the exterior angle is greater than either of the interior and opposite angles.

Note: Originally, my answer quoted the "hyperbolic Pythagorean theorem": If $c$ is the hypotenuse of a hyperbolic right triangle with legs $a$ and $b$, then $\cosh(a) \cosh(b) = \cosh(c)$. But this is way overkill for this problem.