An Investigation of Morley's Theorem
by Amy Benson
Morley's Theorem requires trisecting the angles of a triangle, a task we know to be impossible using traditional compass and straightedge constructions. However, Geometer's Sketchpad will allow us to trisect angles through a series of rotations.
First, I must draw a triangle. I will use rays for reasons that will soon become apparent.
Now, I must measure angle BAC. Then, I will use the GSP calculator to trisect the angle measurement. Next, I mark angle BAC as the center of rotation by double clicking on point A. Additionally, I need to select my trisected angle measurement and "Mark angle measurement" under the transform menu. Now, I select ray AB and rotate it by the marked angle. I follow this step by rotating the resulting ray by the same marked angle, thus trisecting angle BAC.
I must repeat this process for angle BCA and angle ABC.
Having trisected each angle in the triangle, I will now focus on discovery Morley's Theorem. I know that the theorem claims that three of the points of intersection of these angle trisectors are the vertices of an equilateral triangle. The question becomes is this theorem true and if so, for what points of intersection is it true.
After many attempts, I found three points of intersection were, indeed, the vertices of an equilateral triangle (shown here in pink).
Now, the question becomes how to describe these points in a clear, concise manner. I see that points L and K are points on the trisectors of angle C, points K and P are on the trisectors of angle B, and that points P and L are on the trisectors of angle A. So, the points lie on adjacent trisectors, but there are other points of intersection on these trisectors, so why are these points special?
We also notice several other intersecting relationships in this figure. For instance, the triangles shaded in the following figure are all isosceles triangles.