First time on Math SE, so any feedback appreciated
I'm working on some generative art that will draw Pinwheel tiles with svgs. To that end I'm building a function that given a set of three points, spits out the seven points that define the triangle's decomposition.
For the life of me I can't remember how to find these points when the triangle isn't aligned to the x,y plane. I'd like to get the coordinates for the given point:
Is the correct approach to do some basic unit circle trig from the top left point to find the x & y distances from that point in a local coordinate system and then transform back to the coordinate system the rest of the points are in?
Thank you.
Given three points $A, B, C \in \mathbb R^2$ such that $$(a^2, b^2, c^2) = (BC^2, CA^2, AB^2) \propto (1, 4, 5)$$ respectively, it is quite straightforward to compute the coordinates of the other vertices in your diagram. Denote the circled point $D$, and let $E$, $F$ be the midpoints of $\overline{DC}$ and $\overline{DA}$, respectively; finally, let $G$ be the midpoint of $\overline{CA}$. Then by similarity of triangles, $$\triangle BDC \sim \triangle BCA$$ and in particular $$\frac{BD}{BC} = \frac{BC}{AB},$$ hence $$BD = \frac{BC^2}{AB} = \frac{BC}{\sqrt{5}}.$$ That implies that $AB = 5BD$. It follows that $$D = \frac{A + 4B}{5},$$ and the rest is straightforward: $$E = \frac{D+C}{2} = \frac{A+4B+5C}{10}, \\ F = \frac{3A+2B}{5}, \\ G = \frac{A+C}{2}.$$
So for example, if $$A = (3 + \sqrt{3}, 3), \\ B = (5/2, 2 + \sqrt{3}/2), \\ C = (3,2),$$ then $$D = \left(\frac{(3 + \sqrt{3})+4(5/2)}{5}, \frac{3 + 4(2+\sqrt{3}/2)}{5}\right) = \left(\frac{13+\sqrt{3}}{5}, \frac{11 + 2\sqrt{3}}{5}\right).$$