I have two entries, although there is a wealth of elementary geometric examples similar to #1 and several alternative proofs of #2.

(1) Pascal's theorem: If H is hexagon whose vertices lie on a conic section Q then the points $A,B,C$ where the pairs of the opposite sides intersect are collinear.

I think that the first proof used Menelaus's criterion of collinearity and required a figure, as well as keeping track of various points and lines in order to use Menelaus's theorem. A beautiful short proof based on Bezout's theorem is in vol 1 of Shafarevich's "Algebraic geometry":

If the sides of H are given by the vanishing of linear forms $l_1,l_2,l_3$ and $m_1,m_2,m_3$ in homogeneous projective coordinates, where $l_i$ is the opposite of $m_i$, then $l_1 l_2 l_3 - \lambda m_1 m_2 m_3$ vanishes at the vertices of $H$ and one more arbitrarily chosen point on Q, for a suitable $\lambda$; since $6+1>2\cdot 3$, by Bezout, the cubic is reducible, so it consists of Q and another component, which is a line passing through $A,B,C$.

(2) Isoperimetric inequality: If a simple closed curve in the plane has length $L$ and bounds the region of area $A$ then $L^2-4\pi A\geq 0$ (with equality only in the case of a circle).

The first proof of the isoperimetric property of the circle was attempted by Jacob Steiner using the "four rod" method (related to "Steiner's symmetrization"), but it proceeded under the assumption that the minimum is attained and so was incomplete. Weierstrass gave the first rigorous proof based on variational calculus and it was painstaking. Adolf Hurwitz found an essentially one-line proof (after all the notation has been set up) that is reproduced in "Einfuhrung in die Differentialgeometrie" by Wilhelm Blaschke (p.33 of 1950 edition):

$$L^2-4\pi A = 2\pi^2 \sum_2^{\infty} \frac{a_k^2+{a_k}^{\prime 2}}{k^2-1}\geq 0.$$

Here $a_k$ and $a_k^{\prime}$ are the Fourier coefficients of the position vector of the curve w.r.t. unit tangent vector.