Tri as I Might
by Paul Godfrey
This exploration looks at various ways to trisect an angle. First we look at using an unmarked straight-edge and compass. We will also look at trisections that can be performed with marked straight-edge and compass. Then we explore using trisectrices to perform the job. Geometer's Sketchpad [1] was used for most of these explorations. For those having GSP, the GSP files can be obtained by clicking on the figure number.
Reading the College Mathematics Journal [2] I noticed an article about trisecting an angle . It talked about something called a trisectrix. A trisectrix is a curve that can help us easily trisect an angle. One example given was a curve with equation
Fig-1
At first, this sounded like a complicated way to trisect an angle. After all, trisecting a line was a simple matter as Fig-2 shows.
Fig-2
Further, we know that given triangle DBC with rays BJ and BK as shown, any line segment parallel to DC with endpoints on rays BD and BC will be trisected by the rays BD, BJ, BK, BC due to the proportionality principle of similar triangles.
Fig-3
So, we reason that the angle we wish to trisect could be trisected using this method. Since the arc on a circle defined by the legs of the angle is the same measure as the angle, we merely need to trisect the arc. However, attempts to use this simple trisection on an angle quickly proved useless as you can see in .
We start with angleABC. Here we take a circle with center at B intersecting the rays from B through A and C which form the angle we're trying to trisect, i.e., angleABC. Connecting the points of intersection with a cord, we trisect the cord and attempt to trisect the angle from that. We immediately see that this does not work. Why?
Fig-4
We observe the fallacy in this argument when we see that the angular distance AC is not measured in a straight line. In fact, it is measured in an arc. This, then, must be the reason we cannot trisect the angle with the straight line proportionality approach. Maybe, trisecting an angle is not so easy after all.
What we need is some way to trisect the arc. We know that an arc is only a portion of a circumference of a circle. So, what we may need to use is another circle somehow related to the original by three. But before we try that, let's see if there are any angles we can trisect. We find that it is easy to trisect a semi-circle.
Fig-5
We also find it easy to trisect a quarter circle
Fig-6
Now, how do we translate this into trisecting any angle? First, let's note that trisecting the semi-circle leads to a definite segmentation of it's diameter into 1:2:1.
Fig-7
This particular segmentation of the diameter is clearly not a trisection of the diameter. But can we translate this into trisection of the original arc? After trying many exhaustive possibilities, I was unable to do so. I'll leave it to the reader to find a way using this trisection. (NOTE: I found the same true for the trisected quarter)
Let's again examine the problem from a different angle. We have an angle we want trisected. We can create a circle about the angle and hence a corresponding arc. If we can trisect the arc, it follows clearly that we can trisect the angle. Perhaps we can do it by finding triangles as in the next example.
Fig-8
Now this works fine as long as we're not to concerned about total accuracy. We merely adjust a triangulated construction until we reach the angle we seek to trisect.
What about something out of this universe, or at least out of normal space? Say, for instance, hyperbolic space. I tried that as shown in the next figure, but with no success. You might want to give it a shot with this spatial construct yourself.
Fig-9
When told that it is impossible to trisect an angle with a straightedge and compass, people often think it is impossible to trisect an angle. That is not true. I have discovered several ways to trisect an angle In fact, the use of triangles to approximate an angle of arc I talked about above is an example of this. Another way using a marked straight edge and compass, goes like this.
1. Select an angle to be trisected, angleABC. 2. Construct a line parallel to BC at point A. 3. Create a circle of radius AB centered at 4. Mark on the straight edge the length between A and B. 5. Take the straightedge and line it up so that one edge is fixed at the point B. 6. Let D be the point of intersection between the line from A parallel to BC. 7. Let E be the point on the newly named line BD that intersects with the circle. 8. Move the marked straightedge until the line BD satisfies the condition AB = ED, that is adjust the marked straightedge until point E and point D coincide with the marks made on the straightedge . 9. Now that BD is found, the angle is trisected, that is 1/3*ANGLE ABC = ANGLE DBE
This is what it looks like.
Fig-10
Other methods include using special curves, like the trisectrix of Maclaurin, we saw earlier. This is how that one is used. First, we plot the curve in GSP. Then we draw the angle we wish to trisect (ABC) with vertex at (2,0) on the axis. We then draw a line segment from the origin (ED) passing through the point where the BA leg of the angle crosses the curve. Then DEA is 1/3 ABC. (or approximately)
fig-11
And finally, three more trisectrix are (1) Nicomedes' conchoid:
fig-12
(2) the limacon of Pascal:
fig-13
and (3)the trisectrix of Catalan:
Fig-14
One of my favorites is this one. IT MAY BE TRUE [4]. Yes, you have to "download it" to see. The GSP construction is relatively simple. Here is a method which will trisect an arbitrary angle
0) Draw an angle BOA
1) Draw a series of concentric arcs whose centers are the vertex of the angle and cross both rays A and B.
2) Adjust the compass to a smaller radius, say radius R. Then make consecutive arcs along the arcs made in step 1). Start point is where the first arc crosses ray A. Draw threemarks on this arc moving toward ray B. Go to the second arc and again make three marks. Eventually you come to two concentric arcs such that the third mark on one arc lies on the interior of the angle while the third mark on the other arc lies on the exterior of the angle. The ray B passes between the two marks
3) Once the two arcs are found, connect the two third marks with a straight line.
4) Set the compass radius to the distance from the angle vertex to the intersection point of ray B and the straight line.
5) Draw an arc centered at the angle vertex with radius from 4).
6) Reset the radius to R (from step 2)). Draw three marks on the new arc. These represent the trisection points. connect them to the angle vertex
This concludes the explorations of trisecting an angle. Once again, we have no startling results. Nevertheless, I hope you've enjoyed the excursion as much as I.
- GSP
- CMJ, 24/8, Nov 93, Jack Eidswick, "Two Trisectrices for the Price of One Rolling Coin."
- Hesse, Bob, 'Angle Trisection', electronic mail, circa 14 Apr 1995.
- Anonymous