I have an equilateral triangle with know height, length, and 3 coordinate pairs (x, y).
Given this information, I want to determine the 3 coordinate pairs for another equilateral triangle of exactly 1/2 the height and length, the base of which will have a midpoint that sits on the top of the original triangle and will be parallel to the base of the original triangle.
See an illustration of the 1/2 scale triangle on top of original triangle:
How can I calculate these new coordinates?
I think the easiest way is to use vectors. Let's say $\vec P_i=(x_i,y_i), i\in(1,2,3)$, and you want the smaller triangle around $\vec P_1$. Let's call $\vec M_1$ the midpoint between $\vec P_2$ and $\vec P_3$, $\vec M_1=\frac12(\vec P_2+\vec P_3)$. Then in the smaller triangle $\vec M_1'=\vec P_1$ (the middle of the line between $\vec P_2'$ and $\vec P_3'$ is at the vertex of the original triangle). You also have that the distance between $\vec P_1'$ and $\vec M_1'$ is half the distance between $\vec P_1$ and $\vec M_1$: $$\vec P_1'-\vec M_1'=\frac12\left(\vec P_1-\vec M_1\right)$$ or $$\vec P_1'=\frac32\vec P_1-\frac14\vec P_2-\frac14\vec P_3$$ Similarly, for $\vec P_2'$ you have $$\vec P_2'-\vec M_1'=\frac12\left(\vec P_2-\vec M_1\right)$$or $$\vec P_2'=\vec P_1+\frac14\vec P_2-\frac14\vec P_3$$ And finally, $$\vec P_3'=\vec P_1-\frac14\vec P_2+\frac14\vec P_3$$