Centers of Triangles

Rozina Essani

9. In the same original triangle, construct the three secondary triangles of Exercises 6, 7, and 8. Construct the circumcircle for each of the secondary triangles. What do you observe? Can you prove your conjecture?

Given any triangle, let us construct its MEDIAL triangle:

Now let us construct its ORTHIC triangle:

Now let us construct a triangle such that its vertices are the midpoints of AB, AC, and AD.

Now that we have all three required triangles we can start drawing their circumcircles.

By doing the construction the first thing that we notice is that for each of the inner triangles’ sides, the perpendicular bisectors meet at a common point.

This will also prove that all three inner triangles have the same circumcircle, i.e. the distance from the veritices of each triangle to the point of intersection of the perpendicular bisectors is equal.