Trisection of angles
Division with compass
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Trisection of angles
Division with compass

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A first thought could be:
"With a good and precision compass we can dominate, section, divide, etc. segments, angles and circumferences".

As we see in the drawing, using the compass we can section, chop and mark angles, segments, rulers, etc.

As an example we have the trisection of a given angle AB, where with an approximate opening of a third of an angle, we are putting the compass at one end, for example A, turning this with support at the "supposed" points c and d.
Since it will not coincide with the first trial in B, then we open or close the compass a little until in this measurement coincides A and B.
* This measurement is very fast, and it takes approximately one minute for us to match point A and point B, and the measurement is finished.
Then we mark the finally correct points c and d, preferably double-marked (from A to B, and from B to A, as seen in the drawing below ) to observe the exact point of interception(*).
(*) The double arc tracing direction to fix the points c and d (from A to B and from B to A) places them correctly and they act in each of them as when obtaining a bisector, and shows us the correct measurement.
The method is simple: what we do with the compass is measure three equal circumference chords (or three equal sides of a supposed inscribed polygon) which produces three equal angles.

In very acute angles, it is better make the measurement on an arc farther from the vertex of the angle, in order to better use the compass.

In all these works to get the proper measurement we can make 2-4 attempts, and about 1 minute, and so, it is not an long measurement, because we quickly match points A and B and you can no longer improve.

To make a good measurement, you need a very sharp compass (the pencil or marker)

P.S: I think not only this method of sectioning angles and segments is easy, but it can be a good exercise for the use of compass in elementary students.

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"Way to do the trisection quickly".
We open the compass an approximate third of the segment or angle to trisect.
Then we make the two movements of the compass for this measurement, and without separating the needle from the compass of the paper (second movement) and on the current opening, we add or substract a third of the rest that lack or surplus for the correct measurement.
And we measure again, until points A and B coincide with the compass.

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Inscribed polygons to the circumference.

The division of segments and angles with compass is very simple and general, and consists of making a single measurement but with several steps depending on the number of parts in which we want to divide the segment or angle.

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In the case of wanting to draw inscribed polygons to the circumference with this method, the first thing we do is draw the starting point A on the circumference.
Then we open the compass with an approximate opening of a part of the division or side of the polygon that we are going to inscribe.
With this measure of the compass, fixed at the starting point A and lifting only one leg of the compass, we begin to rotate it, resting it on the circumference at points B, C, etc. to get back to point A.
If it doesn't match, we open or close the compass and do another test.
If after making the rotation, the compass makes the starting point A coincide with the ending point A, the measurement will be correct and we will point the points B, C ....., to draw the inscribed polygon.

(The marking of the vertices of the inscribed polygon is done as in the segments and angles. For instance the triangle of the drawing.
Once the exact measurement of the side has been obtained with the compass, the compass is fixed at point A, and from here we cut and mark B.
Now fixed the compass in B, mark us C.
And from C we go back and mark us again in A.
And now, to check the accuracy of the measurement, we do it the other way around.
From A we mark C; from C we mark B; and from B mark us in A.
That is, give us one turn marking in one direction, and another turn marking in the opposite direction, just to check the precision of the measurements.
With other inscribed polygons we do the same: one marking in one direction and another in the opposite direction.)

A final consideration to expose would be the following:
If you can make an "ideal measurement" by opening a compass to take the length of a segment in a single measurement, you can also make "an ideal measurement" in a single measurement of two identical steps.

As a simple problem for this method we could put:

Given a segment A-B, divide it into three equal parts, only with the compass, without ruler.
As in other cases, to get the proper measurement we can make 2-4 attempts, and take less than 2 minutes.
(Here we see the last attempt with a correct result where we have made A coincide with B, and where the measurement ends)

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Concept of division of angles and segments:

The division of segments and angles with a compass can be considered similar to the arithmetic division of number:
if, Ac = AB/n, then Ac x n = AB
Where AB and Ac are segments or angles, and n is a natural number.

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Method of sectioning segments and angles without raising the compass of the paper.

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Thanks you friends.