Exercise 27 Page 613

Let's draw the triangle with vertices on the midpoints of the bases of the red, yellow and blue triangle.

This sure does look like an equilateral triangle. But can we be sure of this?

Proving it

We know that the colored triangles are congruent. This means the height of each triangle, marked with a dotted blue line below, will be congruent as well. We notice three triangles when we do this, $△BYO,$ $△YRO$ and $△BRO.$

The red triangle will be equilateral if the following holds true. This is because $\overline{BY},$ $\overline{BR}$ and $\overline{YR}$ are corresponding sides and would therefore be congruent. We can show this by the SAS Congruence Theorem if we prove that the included angle between the legs are congruent. Let's zoom in on the center of the circle. Remember that the vertex angles of all colored triangles are congruent. Also, the heights intersect the midpoints of the bases of the red, yellow and blue triangles which means it bisects their respective vertex angles. As we can see, the vertex angles of $△BYO,$ $△YRO,$ and $△BRO$ consists of three of the colored triangles vertex angles and another two of the colored triangles vertex angles that have been bisected. Since the angle measure is the same they are congruent and thus the following is true. We know the red triangle is an equilateral and therefore also equiangular.