a coordinate proof
A coordinate proof proves a geometric theorem not with the classical compass-and-deduction reasoning of synthetic geometry, but by placing the figure on the Cartesian plane, naming its vertices with coordinates, and grinding through algebra. It is the practical payoff of Descartes' bridge: a statement about shapes becomes a calculation, and what was a clever construction becomes a routine.
The art is almost entirely in the setup: place the figure cleverly so the algebra stays light. Put one vertex at the origin, run a side along the x-axis, and exploit symmetry, so that as many coordinates as possible are 0 or share letters. Then translate the claim into a computation — equal lengths via the distance formula, parallel or perpendicular sides via slopes, a midpoint via the midpoint formula — and verify the equation comes out true. Crucially, to prove a general theorem you must use general coordinates (letters like a, b, c), not specific numbers: showing it for the triangle with vertices (0,0), (4,0), (2,3) proves only that one case, whereas (0,0), (2a,0), (2b,2c) stands for every triangle of that shape.
Coordinate proofs are wonderfully mechanical where synthetic proofs demand insight — the theorem that a triangle's midsegment is parallel to the base and half its length falls out in a few lines. But honesty requires two cautions. First, the figure must be placed without secretly assuming what you intend to prove (do not force a right angle by your axis choice unless it is given). Second, coordinate methods prove metric and incidence facts beautifully but can obscure why a result is true; the synthetic and coordinate viewpoints are partners, each illuminating what the other leaves dark.
Prove the diagonals of a rectangle are equal. Place it as O(0,0), A(a,0), B(a,b), C(0,b). One diagonal is OB with length sqrt(a^2 + b^2); the other is AC with length sqrt((0 - a)^2 + (b - 0)^2) = sqrt(a^2 + b^2). The two lengths are identical for all a, b — the diagonals are equal, proven in three lines.
A clever placement with letters a, b proves the theorem for every rectangle at once.
Use general coordinates (letters), not specific numbers, or you prove only one example rather than the theorem. And place the figure so your axis choice does not quietly assume the very property you are trying to establish.