**Converse of Pythagoras' theorem:** If the lengths of the sides of a triangle $T$ are $a$, $b$, and $c$, and if $a^2+b^2=c^2$, then the triangle is a right triangle and the side opposite to the right angle is the one whose length is $c$.

**Proof:** Construct a line segment $XY$ whose length is $a$. Then construct a line segment $YZ$ whose length is $b$ which is perpendicular to $XY$. By construction, the triangle $XYZ$ is a right triangle, and therefore, by Pythagoras' theorem and because we are assuming that $a^2+b^2=c^2$, the length of $XZ$ is equal to $c$. So, the triangle $XYZ$ is similar to the original triangle $T$. Since the triangle $XYZ$ is a right triangle, then so is $T$.

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What I find peculiar about this proof is the fact that it uses Pythagoras' theorem in order to prove its converse.

It is not the only situation that I am aware of in which this occurs. For instance, there is a [proof of the converse of Ceva's theorem](https://en.wikipedia.org/wiki/Ceva%27s_theorem#Using_triangle_areas) which uses that theorem. But I am not aware of any example outside Euclidean Geometry.

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> Can anyone provide an example of a theorem of the type $A\implies B$ outside Geometry with a proof which uses the fact that $B\implies A$?
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