Is it possible to find the vertices of an equilateral triangle given its center point?

Question

I was wondering how to find the vertices of an equilateral triangle given its center point?

Such as:

        A
        /\
       /  \
      /    \
     /   M  \
  B /________\ C

Provided that AB, AC, BC = x and M = (50,50) and M is the middle of the triangle, I want to find A, B and C.

Thanks.

Answer 1

Let's assume that $x$, the side of the equilateral triangle, is a known positive quantity and that side $BC$ is horizontal (or that point $A$ is directly above point $M$). Let's also assume you are using Cartesian coordinates (where increasing the first coordinate means moving right and increasing the second coordinate means moving up) and that $M$ is the point $(50,50)$.

Then $x$ is the side of the equilateral triangle $ABC$. By simple geometry we know that the altitude is $\frac{\sqrt 3}2x$. Point $M$ is the centroid of the triangle and is on the altitude which is also the median. This means the distance $AM$ is two-thirds the altitude, namely $\frac{\sqrt 3}3x$.

Therefore point $A$ is $\left( 50,50+\frac{\sqrt 3}3x \right)$.

Point $B$ is $\frac{\sqrt 3}2x$ down from $A$ and $\frac x2$ to the left, so point $B$ is $\left( 50-\frac x2,50-\frac{\sqrt 3}6x \right)$, and point $C$ is $\left( 50+\frac x2,50-\frac{\sqrt 3}6x \right)$.

I tested this answer with Geogebra, and it checks.

Answer 2

Note that we are assuming that $BC$ is parallel to the coordinate axis. Now, we can get all the horizontal and vertical distances we need to solve this problem.

Let $P$ be the midpoint of $BC$. Note that we have $PC = BP = x/2$ and $MP = (x/2)/\sqrt{3}$. Finally, $MA = x/\sqrt{3}$. This should be enough to get all the coordinates.

EDIT: A better method, especially if the triangle can be in any orientation, is to first shift everything so that the center is at $(0,0)$. Now, we have that the distance from the center to any point is $R = x/\sqrt{3}$. Now, in complex numbers, we can express the three points as $R \cdot e^{i\theta}, R\cdot e^{i\theta + 2\pi/3}, R\cdot e^{i(\theta - 2\pi/3)}$. This is because each point is $120^\circ$ apart.

Now simplify shift everything back so that the center is $(50,50)$.

Answer 3

Draw the circle of radius $\frac {\sqrt 3}3x$ around $M$. Pick an arbitrary point $A$ on this circle. Then intersect the circle of radius $X$ around $A$ with the first circle to determine $B,C$ as intersection points.

Note that $A$ could be picked anywhere on the circle, hence the result is not unique.