I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a right angle.
I have already marked out the four points that determine the edges of the roughly square rectangle that will contain the grid. I imagine there is some chapter of the elements that would contain a proof that two lines are perpendicular and at right angles to one another.
I imagine myself being able to apply that theorem to each intersection on the grid, piecemeal, in order to 'grow' it, starting at an arbitrary edge, or perhaps starting at all four and working toward the center.
Of course, this is all for fun, and because I love the thinking style of the Elements. But how would I use the book to do that. What process would I use for 'proving' $90^\circ$ perpendicularity, taking each intersection in turn, like an automaton?