Exploration 8: An Investigation of Circumscribed Triangles
by Elizabeth Gieseking
In this exploration, we are examining the properties of a pair of circumscribed triangles. We start with triangle ABC circumscribed by circle O. We then construct the angle bisectors of angles A, B, and C and label the intersection of these rays with the circle L, M, and N, respectively, as shown below.
We can measure these angles in Geometer’s Sketchpad to look at the relationship between them.
We first notice that the angles of the two triangles are not congruent. We will move points B and C to see how this affects the angles on the two triangles.
From this we see that as we decrease angle A, angle L which is constructed from the bisector of angle A increases. Similarly, when angle B increases, angle M decreases. What is the relationship between these angles? If we examine the measurements closely we find that
But why is this true? Because we are looking at triangles inscribed in the same circle, we can examine the corresponding inscribed angles. The measure of an inscribed angle is one half the measure of the arc that it subtends. This implies that angles which subtend the same arc have the same measure. We also know that points L, M, and N were constructed from the angle bisectors of angles A, B, and C. Let us examine the following diagram.
Red angles: because AL is an angle bisector. because they subtend the same arc. because they subtend the same arc.
Blue angles: because BM is an angle bisector. because they subtend the same arc. because they subtend the same arc.
Green angles: because CN is an angle bisector. because they subtend the same arc. because they subtend the same arc.
Now that we see these angles as the sums of two bisected angles, it is easy to see why and
To get our other formulas, we remember that the sum of the angles of any triangle is 180°. Therefore
which can be rearranged to give us
which gives us the second form of the equations.