Find the length of triangle in three intersection circles

Question

There are three circles ($C_1.C_2.C_3$) with radius r and they intersect each other. Suppose that $d_{ij}$ is the distance between $C_i$ and $C_j$.Is there an equation to express the length of triangle in three intersection circles? Thanks in advance!

Answer 1

No answer but an illustration to the question:

As Mark Bennet pointed out: There is a total of six intersection points.

Answer 2

Theoretically, this is possible but is far too tedious. I have to borrow @AxelKemper 's diagram.

WLOG, we can assume that one of the center is at $O(0, 0) = M_3$, say.

Because all the $d_{ij}$ s’ are known, the co-ordinates of the centers of the other two circles can then be found. Therefore, all the circle equations are then known.

Assume that the circles will intersect as shown.

Solving the circle equations in pairs, we can get the co-ordinates of the 6 intersection points (as pointed out earlier).

These points can be classified into 2 types. The three being circled formed the vertices of the required triangle. They have the following characteristics:-

"It must be the point of intersection of two circles and is inside the remaining circle."

Take X23 (let say it is located at $(p, q)$)as an example. It comes from the intersection of the green and the blue circle but lies inside the red circle (let say it equation is $f(x, y) = 0$).

If the “tangent length (to the red circle)” is imaginary, it is inside the red circle. That is, if $f(p, q) < 0$, the point is one of the required point.

After the above classification process, the three vertices of the required triangle are found (with their co-ordinates known). Lengths of the sides of that triangle as well as its perimeter or even its area can be determined.