### TRISECTING AN ANGLE (Added to this website on 3/10/11.)

During a telephone conversation with my brother, he showed me a way that works well to trisect a line.   He also mentioned that trisecting an angle seems to be impossible and that someone had proven that it was impossible.   So apparently, trisecting an angle using only a straightedge, a compass, and a pen or pencil is impossible.   Perhaps that is true, but after I became interested I found that one can trisect an an angle with an error that is something less than .085 degree (less than a tenth of degree), using only those tools mentioned.   If one wants results, this degree of accuracy is more than adequate.   By repeating the procedure with more bisectors, the error can be reduced to virtually nothing - something a computer can do easily.   It was my brother who was responsible for showing me a method that is "out of the box" by mentioning the way a line can be trisected.

For angles of 60 degrees or less, the procedure is relatively simple.   For angles greater than 60 degrees, the procedure is basically the same but with more of the same procedure. For those who are interested, this is how it is done:

1.   Draw an angle of 60 degrees or less on a piece of 8.5x11 paper.   Make each side in excess of 7 inches so you will have room for what follows.   Label the intersection of the lines "In".   Label the left line "L" and the right line "R".

2.   Using your compass, make a mark at a point between 1.25 and 1.5 inches from "In" on "R".   Call that mark "point 1".

3.   Keep your compass set for the same distance you chose and make other marks on "R" at twice, thrice, and four times that distance from "In".   Call the one at twice the distance "point 2", the one at thrice the distance "point 3", and the one at thrice the distance "point 4".

4.   Make three arcs across the angle, passing through points 2, 3, and 4.   Don't make any arc through "point 1".   Call the arcs 2, 3, and 4.

5.   Using arc 4 where it crosses "L" and "R", bisect the angle.   Call the bisecting line "S".

6.   Using the same arc where it crosses "L", "S", and "R", make two other bisectors, one between "L" and "S", and the other between "S" and "R".   Call them "Q1" and "Q2".

7.   Make "A" the point where "Q1" crosses "arc 4", "B" the point where "Q2" crosses "arc 4", and "C" the point where "S" crosses "arc 2".

8.   Draw a line from "A" to "C", and another line from "B" to "C".   Make "D" the point where line AC crosses "arc 3", and make "E" the point where line BC crosses "arc 3".

9.   Draw a line from "In" through "D" and another line from "In" through "E".   These are the trisecting lines.

When the angle to be trisected is between 60 and 120 degrees, it is necessary to use the same procedure but with two other bisectors between "L" and "Q1", "Q1" and "S", "S" and "Q2", and "Q2 and "R".   If you are good with your drawing, you only need to make one more bisector -- the one between "Q2" and "R".   Where this bisector (call it "T") crosses "arc 4" you have point "F".   Where it crosses "arc 2" you have point "G".   Make a line from "F" to "G". FG crosses "arc 3" at point "H".   After the new line crosses "arc 3", you may use twice the distance along "arc 3" from "R" to "T" to determine points on "arc 3" for the trisecting lines.

When working with angles between 120 and 180 degrees, another set of bisectors must be added so as to keep the lowest distance between bisectors 15 degrees or less.   The margin of error in this method was determined as follows.

1.   Find the difference between the arc distance and the chord for 15 degrees.

2.   Notice that the error is going to be no more than a quarter of that difference at worst.

3.   Convert the answer to degrees.

The chord is found using the sine of 15 degrees times the radius.   There are 24 fifteen-degree angles in 360 degrees.   Divide twice the radius times pi by 24 for the arc distance.   Subtract the chord from the arc distance and divide the answer by 4.   Convert this to degrees by dividing it into the circumference and then dividing that into 360 degrees.

For More Accuracy
1.   Leave out bisector "Q1".
2.   Measure the straight-line distance between "C" and "point 2" with your compass.
3.   Draw a separate line.   Call it "L".
4.   From one end of "L" lay out the straight-line distance from "C" to "point 2".
5.   Call the above distance "chord 2".
6.   On "L" add the straight-line distance from "B" to "point 4".
7.   Call the above distance "chord 4".
8.   Call the total of the two distances "T".
9.   Bisect "T".   Call one-half of "T", "chord 3".
10.   Use your compass to lay out "chord 3" from "point 3" to "E".
11.   From "E", use the same "chord 3" distance to go from "E" to "D".

The degree of accuracy of the foregoing is .0021462 times the number of degrees in one-third of the angle being trisected.   For instance, for an angle of 60 degrees to be trisected, one-third is 20 degrees.   20 times .0021462 is .041198 degree - less than half of tenth of a degree.

Easier but Less Accurate
1.   Leave out "point A", "arc 4", and "Q1".
2.   Measure the straight-line distance between "C" and "point 2" with your compass.
3.   Call the above distance "chord 2".
4.   Use your compass to lay out the length of "chord 2" from "point 3" to "E".
5.   From "E", use the same "chord 2" distance to go from "E" to "D".

The degree of accuracy in this method is .0066129 times the number of degrees in one-third of the angle being trisected.   For instance, for an angle of 60 degrees to be trisected, one-third is 20 degrees.   20 times .0066129 is .126297 degrees - slightly over a tenth of a degree.