Assignment 8

Patricia Johnson


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


Construct the diagram using GSP.

From the measurements above can you find a relationship between angles L, M, and N in terms of angles A, B , and C?

Given that LA bisects angle A, BM bisects angle B, and NC bisects angle C show that :

Angle M = 1/2 Angle A + 1/2 Angle C

Consider the diagrams:

We see in diagram 1 that part of Angle M = 1/2 Angle A since they are both subtended on the chord BL. In diagram 2, we see that the rest of Angle M = 1/2 Angle C (both angles stand on the chord BN). We can, therefore, conclude that

Angle M = 1/2 Angle A + 1/2 Angle C

Construct the chord LC. Identify and shade the two triangles that stand on the chord LC.

Find the two angles subtended on this chord.

Repeat for chord MC.

What is the relationship between angle N and angles A and B?

Now show that angle L = 1/2 Angle B + 1/2 Angle C.


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