# An equilateral triangle inscribed in an ellipse

Clear["'.*"]
{ecc, p, r1, t1} = {0.35, 1., 1.4, 0.8}
EQ = {p/r2 == 1 - ecc Cos[t2], p/r3 == 1 - ecc Cos[t3],
r1^2 + r2^2 - 2 r1 r2 Cos[t1 - t2] ==
r3^2 + r2^2 - 2 r3 r2 Cos[t3 - t2],
r3^2 + r1^2 - 2 r3 r1 Cos[t3 - t1] ==
r3^2 + r2^2 - 2 r3 r2 Cos[t3 - t2]}
NSolve[EQ, {r2, r3, t2, t3}]


First radius vector is given/defined in the first line. Also given are ellipse properties ( $ecc, p =$ eccentricity and semi-latus rectum).

I am trying to inscribe an equilateral triangle in order to find two more vector arms $r2,r3$ along with their polar angles ${t2,t3}.$

What assumptions or essential changes did I miss that makes this to hang? Is there a version problem?

• I suspect I know what ecc and your other variables represent, but you might want to indicate what they're supposed to be for reference. See this as well. Mar 30 '18 at 15:21
• Yes sir, classical symbols of Newton planetary motion focal conic used. Mar 30 '18 at 19:51
• Perhaps you did not see the image I just deleted or gone through the program. There are three radial arms. First one has length $1.4.$ Other two $(r2,r3)$ are unknown at start of computation. It would be also evident from the program. Feel free to say if my problem expression is in some way lacking. Mar 31 '18 at 17:46
• Because I chose origin / ellipse focus as reference of vectors starting , not its center of gravity. The link has an image of sorts. Mar 31 '18 at 18:39

NMinimize[{1, EQ}, {r2, r3, t2, t3}]

• you might try FindInstance as well.. Mar 30 '18 at 21:27