I'm doing some generative design for architecture and I can't quite get my high school geometry over the line on this one. I have filled pages with pythagoras; I must be ignorant of a more powerful method:
I know all four lengths of a convex quadrilateral, AB, BC, CD, DE. I know the coordinates of A and B I know that CD is flat ie. it has a normalized vector of (1,0,0) I need to find the coordinates of C and D (they share Y values)
(I'm aware that there are two solutions either side of AB, but it will be trivial to know which one I want)
Let $B'(X_B + CD, Y_B)$ be the point obtained by translating $B$ to the right by a distance equal to the (known) distance $CD$. Because the same translation takes $C$ to $D$, we know that $B'D = BC$.
Therefore $D$ must be the point at distance $AD$ from $A$ and at distance $BC$ from $B'$: it is the intersection of two circles whose center and radius we know.
Solving for that intersection point can boil down to an equation in coordinates, starting from the equation of a circle with radius $r$ and center $(h,k)$: $$(x-h)^2 + (y-k)^2 = r^2.$$ You can find discussions of that problem here for example. Briefly, subtracting the equations of the two circles gives a linear equation, which you can solve for $y$ in terms of $x$ (or vice versa), and then substitute into one of the circle equations to get a quadratic equation.
Depending on your situation, you might have other ways to find the intersection of two circles, such as by using a compass to draw the circles and mark the intersection :)