**An Initial Construction**

To begin this investigation, construct Triangle ABC, it's circumcenter and circumcircle. Next, construct the angle bisectors of Angles A, B, and C and the point where each bisector intersects the circumcircle. Finally, construct a triangle from these points.

Are there any relationships between the
angles in **Triangles ABC** and **LMN**?

**Angle Relationship**

To describe one relationship between
the angles of **Triangles ABC** and **LMN**, one must establish
the relative positions of the angles. If you were to trace along
the circumcircle from **L** in either direction, the first
points you will arrive at are **B** (if you trace counterclockwise)
and **C** (clockwise). We will describe this relationship of
**Angles L**, **B**, and **C** by saying that **Angle
L** is between **Angles B** and **C**. By similar logic,
**Angle M** is between** Angles A** and **C**, while
**Angle N** is between **Angles A** and **B**. The following
conjecture is based upon this definition:

**For any two angles in Triangle
ABC, the average of their measures is equal to the measure of
the angle of Triangle LMN that is between them.**

Some quick measurements with Geometer's Sketchpad confirms this. Click Here to open a sketch showing these measures.

**A Proof**

The proof of this conjecture uses the property that states: The measure of an inscribed angle is half the measure of its intercepted arc. Refer to the sketch below.

For each of the arcs listed below, assume that the minor arc (less than 180 degrees is being described).

The proof begins by showing that the
measure of **Angle MLN** (previously called **Angle L**)
is the average of the measures of **Angles ABC** (**Angle
B**) and **ACB** (**Angle C**). Since **Angle ALN**
and **Angle ACN** intercept the same arc, they must be congruent.
In a similar manner, **Angle ALM** is congruent to **Angle
ABM**. By the angle sum property,

**m Angle MLN = m Angle ALN + m Angle
ALM**

Substituting in the congruent angles,

**m Angle MLN = m Angle ACN + m Angle
ABM**

Since we know that **Rays CN** and
**BM** are bisectors, then the measure of **Angle ACN**
is half the measure of **Angle ACB** and the measure of **Angle
ABM** is half the measure of **Angle ABC**. Thus,

**m Angle MLN = 0.5(m Angle ACB + m
Angle ABC)**

It is indeed the average. A similar proof can be constructed for the other two cases. QED : )

**Extension**

Construct another inscribed triangle
using the process above and **Triangle LMN** as the starting
triangle. If this process is repeated through many iterations,
what would happen to the each newly constructed triangle and its
angle measures? The spreadsheet below may give a clue. Can you
prove your conjecture?