An Exploration of the Angle Bisectors of an Acute Triangle

By: Sydney Roberts

 


 

Consider any acute triangle ABC and its circumcircle.

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Now consider the angle bisector of each angle within the triangle, and label the points where these bisectors will intersect the circumcircle as points L, M, and N respectively.

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Now we want to find angles L, M, and N in terms of the angles A, B, and C. However, notice that all of the angles (A, B, C, L, M, and N) are inscribed angles. Hence, the inscribed angle theorem holds which says that the measure of the intercepted arc is exactly twice of the measure of the inscribed angle. A consequence of this is that the property holds for all inscribed angles of an intercepted arc. For example, the measure of arc BC is the same as twice the measure of and .

Therefore, to find angles L, M, and N in terms of A, B, and C we will use this fact. Start by considering angle L and arc AM. Notice that since ray M is the angle bisector for angle B, then . However, note that arc AM also has another inscribed angle, . Hence, because of the previously stated theorem, . Hence, .

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Now, since ray CN is the angle bisector of angle C, then has an intercepted arc AN which has another inscribed angle, . Hence, .

And therefore, . Hence, we have described angle L in terms of angles A, B, and C as desired. Now, letís do the same for angle M.

Start by considering and arc BL. Since ray AL is the angle bisector of angle A, then we know . The intercepted arc here is arc BL, so we also know that is half the measure of the intercepted arc. Again, there is another intercepted arc to consider and that is Hence, .

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Now consider and intercepted arc NB. Since ray CN is the angle bisector of angle C, we know that which is also equivalent to Ĺ the measure of the intercepted arc NB. Therefore, we can look for other inscribed angles that intercept the same arc and we see that does. Therefore,

 

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Therefore, which is in terms of our original angles, as desired.

Finally, we can do the same thing for angle N and we see that since ray AL is the angle bisector of angle A, then . Now considering the intercepting arc LC, we see that is also the same as . Hence, .

Now consider . Since ray BM is the angle bisector of angle B, then which is also half of the measure of the intercepted arc CM. However, this arc has another inscribed angle, Therefore, .

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Hence, . In conclusion we can define angles L, M, and N in terms of A, B and C since

To determine whether or not you think these defintions only hold for acute triangles, use the following GSP file to make your own conjectures. Angle Bisectors

Hopefully you can conclude that this will always hold since our definitions are made by using arbitrary arc and angle measures.


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