mathematics education - Theorems in Euclidean geometry with attractive proofs using more advanced methods

A nice example is Pascal's theorem for the circle:

If a hexagon is inscribed in a circle then the intersections of opposite
sides are collinear.

Plücker gave an elegant proof of Pascal's theorem as a consequence of
Bézout's theorem. View the unions of alternate sides of the hexagon as
cubic curves

$l_{135}=0$ and $l_{246}=0$.

They meet in 9 points, 6 of which are the vertices on the circle $c=0$. But we
can choose constants $a,b$ so that the cubic

$al_{135}+bl_{246}=0$

passes through any point. Taking this point on the circle, the circle and the cubic have at least 7 points in common. By Bézout's theorem, the curves have a common component, necessarily the circle $c=0$, since $c$ is irreducible.

Hence $al_{135}+bl_{246}=cp$, for some polynomial $p$, which must be
linear. Since $al_{135}+bl_{246}=0$ contains all 9 points common to
$l_{135}=0$ and $l_{246}=0$, while $c=0$ contains only 6, the remaining 3
(intersections of opposite sides of the hexagon) must lie on the line $p=0$.