*The theorem that the general angle cannot be trisected with ruler and compass alone is true only when the ruler is regarded as an instrument for drawing a straight line through any two given points and nothing else.*

By permitting other uses of the ruler the totality of possible constructions may be greatly extended. The following method for trisecting the angle, found in the works of Archimedes, is a good example.

**METHOD:**

Figure 1 : Archimedes’ trisection of an angle

[pp. 138, What is Mathematics?, © Oxford University Press Inc. ]

__STEP – 1__ Let an arbitrary angle x be given, as in Fig. 1.

__STEP – 2__ Extend the base of the angle to the left, and swing a semicircle with O as center and arbitrary radius r.

__STEP – 3__ Mark two points A and B on the edge of the ruler such that AB = r. (use compass to measure, thus using ruler to measure!!)

__STEP – 4__ Keeping the point B on the semicircle, slide the ruler into the position where A lies on the extended base of the angle x, while the edge of the ruler passes through the intersection of the terminal side of the angle x with the semicircle about O.

__STEP – 5__ With the ruler in this position draw a straight line, making an angle y with the extended base of the original angle x.

**RESULT: **This construction actually yields

**PROOF:**

Figure 2: Labeling Archimedes’ trisection of an angle

[pp. 138, What is Mathematics?, © Oxford University Press Inc. ]

As in Fig 2,

, and

Similarly in , , and

From , we have,

Similarly, from, we have,

Now using in , we can eliminate to get:

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I wish someone told me this trick when I was in X….

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