In this tutorial, we use GeoGebra
to explore the minimum sum of the distances of a point on the interior of a
triangle to its sides. We learn how to use the *Point on
Object *tool and the *Perpendicular Line* tool. We also learn how to compute using
the Input bar. The final output of this tutorial is shown in the following
applet.

Steps in Creating the Applet

Click the check boxes and move the vertices of the triangle to explore. What do you observe?

In any triangle, a circle called *circumcircle *maybe drawn that passes through its vertices. The center of the circumcircle is called the *circumcenter*. The perpendicular bisector of the three sides of any triangle passes through its circumcenter.

In the applet below, *ABC* is a triangle whose sides are also sides of equilateral triangles that contain the green line segments. Points *P*, *Q*, and *R* are centroids of the equilateral triangles.

1) What do you observe about the figure?
2) Move points *A*, *B*, and *C*. Are your observations still the same?
The Napoleon theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those triangles themselves form an equilateral triangle.