## The internal angle bisectors of triangle ABC are extended to meet the circumcircle at points L, M, and N, respectively. Find the angles of triangle LMN in terms of the angles A, B, and C.

## Does your result hold only for acute triangles?

Before we start to define the angles of triangle LMN in terms of the angles A, B, and C, we will first consider what we are looking at and what we will need to use. We have two triangles whose vertices lie on the circumcircle of triangle ABC. Because the circumcircle of triangle ABC goes through the vertices of the triangle and triangle LMN was defined by the intersection of angle bisectors and the circumcircle, we are working with inscribed angles. Let us review inscribed angles. We discussed inscribed angles in the context of the football field problem in assignment 5. Inscribed angles are angles that are formed by two chords with a common endpoint, the vertex of the angle, on the circumference of the circle. The measure of this angle is half of the arc that the angle intercepts. For example, in the diagram above, the measure of angle ABC is half of the measure of arc AMC or the measure of arc AMC is twice the measure of angle ABC. In order to complete this problem and define the angles N, M, and L in terms of angles A, B, and C, this idea will be used repeatedly. Moreover, since we took the angle bisectors of angles A, B, and C, we also bisected the intercepted arcs of those angles. So, for example, the measure of the intercepted arc of angle BNC is half the measure of the intercepted arc of angle C, arc BNA.

Lastly, as long as two inscribed angles share the same intercepted arc, the angles are equal.

Below is a diagram with the angles measured by the capabilities of the GSP software. I placed the measure of each angle on the arc that that angle intercepts. So the measure of the intercepted arc will be twice the measure of the inscribed angle. Using this diagram and the labeling of it, we are able to see how to define the angles L, M, and N in terms of angles A, B, and C.

Let's first look at angle A. Again, the blue dotted lines are angle bisectors so angle CAL is congruent to angle LAB. But the intercepted arc of angle CAL is the same as the intercepted arc of angle CNL. And the intercepted arc of angle LAB is the same as the intercepted arc of LMN. And as we talked about earlier, since the intercepted arcs are the same, these angles must be the same. This can be repeated multiple times for all the different angles and intercepted arcs.

So we get that A + B = N.

And A + C = M.

And B + C = L.

But does this only hold for acute triangles??

Here is an example with an obtuse triangle.

And an example with a right isosceles triangle.

In each of these cases, our relationship between angles A, B, and C and angles L, M, and N still holds true.

A + B = N

A + C = M

B + C = L

Note the relationship between the algebraic and geometric representations. The angles at the intersection of the angle bisectors and the circumcircle (angles L, M, and N) are found by taking half of the angles on each side of the that angle.