2
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ 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
3
$\begingroup$

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.