Megan Langford

We will first take a look at an image of a triangle and its circumcircle. We will label the vertices of the triangle A, B, and C, and we will label the circumcenter of the triangle X.

Now, let’s construct angle bisector for each angle of the triangle.

Next, we will mark the points where the angle bisector lines intersect with the circumcircle. We can then connect these lines to form another triangle, which we will label LMN.

This is the construction we will analyze. Our goal is to determine the angle measurements of triangle LMN in terms of the angles in triangle ABC.

The first thing we will do is examine the figure at hand. What do we immediately know about any of the angles in the picture?

Analyzing the Original Triangle’s Angles

Well, since we know that the dashed lines are angle bisectors, this means each of them cuts the original angle measurement in half.

For example, the measurement of will be exactly half the measurement of .

Additionally, geometry tells us that each of the vertices of the triangles represents an inscribed angle of the circle, since each of them are formed by 2 chords that share an endpoint. Therefore, we know that the length of the arc representing the portion of the circle that is in the interior of the inscribed angle is exactly twice that of the inscribed angle measurement.

For example, the measurement of the arc formed by points C and B is exactly twice the measurement of .

Putting these two concepts together, we know that , and then the arc formed by points C and L on the circle is the same as the measurement of . Therefore, the arc formed by points C and L on the circle is equivalent to .

Applying this to all the vertices of the original triangle, we have

CL=

BN=

AM=

Analyzing the Red Triangle’s Angles

Now that we know more about the original triangle, let’s examine the relationships for the red triangle. Similar to our analysis of the original triangle, we can see that

ML=

ML=MC+CL

=MC+CL

Since we know that MC=AM because both arcs were formed by the bisection of an angle, we can use the information we computed above to show that

Hence,

This relationship will hold for all sides of the red triangle, giving us

Does This Hold for Triangles that are not Acute?

Let’s try this solution with an obtuse triangle. We will have Geometer’s Sketchpad calculate the angle measurements, and then we will check that our solution still holds.

Indeed, this is consistent with our solution, and the result will work for any of the angles on our red triangle.