Trisection of angles
Division with compass
Of ferman: Fernando Mancebo Rodriguez--- Personal page. ----Spanish pages
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Trisection of angles
Division with compass
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.
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.
"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.
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.
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.
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)
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.
Thanks you friends.