Euclid's Proof of Pythagorean Theorem. Can you explain this one step?

Question

How is half of square CFEG, that is triangle FGC is equal to triangle FCD?

I was explained that it's because the base of the triangles are the same, that is CF.

But they don't have the same height? HOW?

edit: I made DGE collinear.

Still doesn't make sense to me. How is GC the height of triangle GCB?

Answer 1

They actually do have the same height, if the height is measured as the altitude on $\overline{CF}$. The reason why you're not seeing it is because the drawing is inaccurate: in fact, points $D$, $G$, and $E$ are collinear if $\angle DGC$ is a right angle. Since $D$, $G$, $E$ all lie on a common line, and this line is parallel to $\overline{CF}$, as the opposite sides of a square are parallel, it immediately follows that $$|\triangle DCF| = |\triangle GCF|,$$ since they share the same base $CF$, and the height is equal to the length of $GC$.

Answer 2

In that drawing, you'd never think that $DGE$ was a line!

$DCF$ and $GCF$ have the same base and same height as $DG$ is parallel to $CF$.