If I have an equilateral triangle (the black one), with a known circumscribed circle radius R = h1, and a known angle θ, how can I find the value h2, which is the R for the outer equilateral triangle with edges that intersect with the points of the inner triangle?

I'm using this for a generative art project where I want a sequence of nested triangles.

nested equilateral triangles

  • $\begingroup$ Familiar with locus of points from which a given segment is viewed at a given angle? $\endgroup$ Jan 7 '21 at 1:11
  • $\begingroup$ @AlexeyBurdin I'm not sure I understand the question. I looked up locuses and they seem like they would be useful, but I'm not sure how to use them in equations. $\endgroup$
    – emma
    Jan 7 '21 at 1:28

enter image description here

Use the sine rule to set up

$$\frac{x_1}{h_1}=\frac{\sin\theta}{\sin 30},\>\>\>\>\> \frac{x_2}{h_1}=\frac{\sin(120-\theta)}{\sin 30} $$

$$\implies x_1+x_2 = \sqrt3h_2 = 2h_1\sin\theta+ 2h_1\sin(120-\theta)$$

which yields

$$h_2 = h_1(\sqrt3\sin\theta+\cos\theta)$$


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