How can I work out the length of the sides of an Equilateral triangle that exists in a circle?

Question

I'm making a game and need a bit of mathematics help.

If I have a circle with a radius of $x$ and there is an equilateral triangle in the center made up of $3$ equal chords. How can I work out the length of the sides of the triangle?

I actually want to split the equilateral triangle into $3$ different isosceles triangles(making a straight line from the center to the corner of the equilateral triangle). The $2$ equal sides are obviously $x$ in length. But the last side's length eludes me.

I'd prefer not to use trig. Computers generally don't like any form of roots. But I can probably find a way to optimize it if that's my only solution.

Thanks in advance.

Triangle I added this Image for clarity. I need the value y.

Answer 1

No trig is needed for the answer, though the law of Cosines gives a direct approach.

To do it via basic geometry, draw a radius out to one of the vertices. By Symmetry it bisects the angle there. Dropping the perpendicular out to a side of the equilateral triangle, gives us a $30-60-90$ right triangle with hypotenuse $x$, the radius of the circle. Of course, the side opposite the $60$ degree angle has length $\frac {\sqrt 3}2\,x$ from which it follows that the length of one side of the equilateral triangle is $$\boxed {\sqrt 3\,x}$$

Answer 2

This is what I used to do before I came to know about trigonometry. Let $BD$ be the median. Medians of an equilateral triangle pass through the circumcenter. So circumcenter is the centroid of triangle. Centroid divide median in $\frac21$. So $\frac{BO}{OD}=\frac21 \implies BD=\frac{3x}{2}$. $BD$ is also the altitude of the triangle. Now altitude of an equilateral triangle of side $a$ is equal to $\frac{a\sqrt3}{2}$. $$\frac{a\sqrt3}{2}=\frac{3x}{2}\\ a=\frac{3x}{\sqrt3}=\sqrt3x$$