TITLE: Prove the statement 'World is not flat'
QUESTION [7 upvotes]: In my graph theory book exercise, I found a problem that:

Prove that the World is not flat using Mathematics


This picture is used in the exercise, but no idea of applying it.
I would have used the angles don't add to $180^\circ$, but the author is using a fourth city.

REPLY [5 votes]: The two diagonals $p$ and $q$ of a plane quadrilateral and the four side lengths $a$, $b$, $c$, $d$ are related by the Cayley-Menger determinant: $$\det\pmatrix{0&a^2&p^2&d^2&1\cr a^2&0&b^2&q^2&1\cr p^2&b^2&0&c^2&1\cr d^2&q^2&c^2&0&1\cr1&1&1&1&0\cr}=0$$ See https://en.wikipedia.org/wiki/Quadrilateral#Properties_of_the_diagonals_in_some_quadrilaterals