Theorems in Euclidean geometry with attractive proofs using more advanced methods The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The Mathematician's Brain or Shifman's book You Failed Your Math Test, Comrade Einstein.
Are there other good examples of simply stated theorems in Euclidean geometry that have surprising, elegant proofs using more advanced concepts?  Such examples are valuable pedagogically since they illustrate the power of the advanced methods.
 A: 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 $p=0$.
A: What about the proof of the isoperimetric inequality using Fourier analysis?
A: I like the following result which plays a role in Marden's Theorem:
Given any triangle, there is an inscribed ellipse which meets the midpoints of all three edges.
Proving this directly is rather difficult (in fact, I'm not sure how to), but it is very easy to do if you know anything about linear transformations.
A: A very nice example is given by the Villarceau circles: a revolution torus is cut by a bitangent hyperplane along the union of two circles. You can of course make the computation, but when you know some projective algebraic geometry you can prove it in a few words. Roughly:


*

*The revolution torus has an algebraic equation of degree four, so that it intersects any plane along a degree four curve.

*If the plane is bitangent, then this curve has two double points so that it must be the union of two ellipses.

*It is easily checked that in the complex world, the torus as well as the plane contain the circular points at infinity, so that in fact the ellipses are circles.
A: Take a triangle with a circle $\Gamma_0$ tangent to two of three sides (you may also think that sides of the triangle are made out of circle arcs).
Construct a chain of circles $\Gamma_1,\Gamma_2,\dots$ on such a way that $\Gamma_{n+1}\not=\Gamma_{n-1}$ is tangent to $\Gamma_n$ and two of the sides of triangle. 
Prove that 
$$\Gamma_6=\Gamma_0.$$
I do not know the proof, but I was told that it is hard to do without knowing elliptic functions.
P.S. I do not know the references --- please feel free to add it :)
A: The elementary proof of Morleys theorem is rather incomprehensible compared to the non elementary ones.
A: Somebody has this as a hobby.
A: The "most elementary theorem of Euclidean geometry" as proved in http://www.math.psu.edu/tabachni/prints/grid.pdf (p.9). The Poncelet's Porism in Gjergji Zaimi's answer is also proved there.
A: The classification of conics would be an example, but I don't know if you count matrix reduction as a "more advanced concept".
A: I found the introduction to automated theorem proving using the Ritt-Wu method of characteristic sets in Cox, Little, and O'Shea to be quite enlightening for pedagogy on introductory algebraic geometry.
I don't know if any such theorem proven by the Ritt-Wu method "surprising" but it was a great way to tie a lot of the ideas in CLO's previous chapters.
Chou and Gao describe proving Pascal's theorem, the Butterfly theorem, Morley's theorem, etc. using the Ritt-Wu method.
A: (Edited after Willie's comment)
Many theorems about a triangle can be proved easily using the fact that all triangles are linearly equivalent. One example is that the segments that join each vertex to the midpoint of the opposite side intersect in a single point.
(This actually appeared on a written qualifying exam I took as a graduate student, and I did in fact use the linear algebra approach, since there was no chance I could recall the Euclidean geometric proof.)
A: Let's see. Most of the results of projective geometry are much harder to show by more elementary methods. I think the reason for this is that they rest more on the incidence axioms while the elementary methods play more with the metric properties. 
An interesting question that arises for any of these examples is to detect why is that the case that the elementary proofs are harder.
A classical example is the constructibility of regular polyhedral with ruler and compass. 
