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.
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