[alogo] Division problem

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The above two pictures give the solution for the triangle and the case N=3 of the general division problem:
Divide a polygon through a succession of N points on its perimeter, so that successive points are at a fixed distance or, in other words, the polygon with vertices the dividing points is equilateral and inscribed in the polygon. In this and several files, referred below, we handle the cases of triangles and N <= 6. The general problem will be handled in subsequent files (under preparation).
Related to this division problem is also the problem of finding the inscribed equilateral polygon of N sides with minimum/maximum area/perimeter. Here e.g. the minimum inscribed equilateral does not exist, if we allow figures like the above on the left. If we restrict ourselves to equilateral inscribed as in the case of the second figure, then there is a minmal one.

Info on animation: Points E in the first and E, D in the second figure are modifiable through the tool [Select on contour] (ctrl+2). Moving them changes the location of the inscribed quadrilateral.
The subject handled here is a particular one of a more general problem, further investigated in the Tetradivision.html .

Look at the file Pentadivision.html for a discussion on the corresponding problem for inscribed pentagons.

With growing N there is a variety of cases of equilateral polygons inscribed in a triangle. A first classification of these cases happens by considering the tripples (a, b, c) of integers with a+b+c=N. Each such tripple represents the distribution of vertices on each side of the triangle: a vertices on the first side, b on the second, etc. Some tripples may be empty. e.g. (1,1,4) for hexagons is impossible, since three sides, forming a broken line should have equal length with a line segment with the same end-points. Look at the file Hexadivision.html for a discussion of the case of inscribed hexagons. In the last mentioned document appears a general method to handle the problem, using the angles of the enclosing polygon and some additional angles defining the slope of some sides of the inscribed equilateral with respect to some sides of the enclosing polygon. All these ideas can be generalized for equilateral N-gons inscribed in other M-gons. The various classes of inscribed N-gons are, initially, distinguished by the partitions a+b+...+z = N, where in the sum appear M terms. After the investigation of these, admittedly many, one should consider the cases, where some vertices of the inscribed coincide with some vertices of the enclosing polygon.

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