- What shape do the contours of the same color make?
- What is the point at the "center" of the triangle has the deepest shade of violet?

*foci*. We can rephrase the questions above without referencing color as, "Describe the set of all $P$ for which the sum $PA + PB + PC$ is a fixed value $d$" and "For which point $P$ is the same sum minimized?" If we consider diagrams with only one or two foci, the questions have well-established answers. With one focus, the contours form circles and the "center" point is the center of the circle. With two foci, the contours form ellipses, and the "center" points form the segment connecting the foci.

**Investigation**

- Each contour is a convex figure.
- A contour may enclose zero, one, two, or three of the foci. Note that this is in contrast to the case of a circle or an ellipse.

**Hypothesis**

*tri-center*of the points $A, B,$ and $C.$

**Case 1**: The three foci form a triangle $\Delta ABC$ all of whose angles measure less than $120^{\circ}.$ In this case the point $P$ is the "120°-point" of the triangle as illustrated in the image. The 120°-point of a triangle $\Delta ABC$ is the point $P$ such that $m\angle APC = m \angle BPC = m \angle CPA=120^{\circ}.$

**Case 2**: The three foci form a triangle $\Delta ABC$ one of whose angles measures greater than $120^{\circ}.$ In this case, the point $P$ is located at the vertex of this largest angle.

**Proof**

To aid our proof, we note that minimizing $AP' + BP' + CP'$ is equivalent to minimizing $BP' + CP' - P'M$ and divide our proof into two cases.

Let $x=PM, x\sqrt{3} = BM = CM, 2x=PB = PC.$ In addition, let $y=BP' = CP'$ and $z=PP'.$

**Case 1**: $P'$ is above $P.$ To show our point, we need to show that $3x < 2y-x-z,$ which is equivalent to $z < 2y - 4x.$

By the Pythagorean Theorem, $z=\sqrt{y^2-3x^2}-x.$ We show the result by reversing the steps of

$$\sqrt{y^2-3x^2}-x<2y-4x$$

$$\sqrt{y^2-3x^2}<2y-3x$$

$$y^2-3x^2<4y^2-12xy+9x^2$$

$$0<3y^2-12xy+12x^2$$

$$0<3(y-2x)^2,$$

which is certainly true, because $2y-3x$ is positive.

**Questions**

- Can I provide a description of the shape of the contours generated by three foci?
- Can I prove that contours are always convex?
- Can I complete the above proof for non-isosceles triangles?
- I was able to provide a simple description of the tri-center in terms of the 120°-point of a triangle. Does this generalize easily to the case with four or more foci?
- How can one construct the 120°-point of a given triangle?