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 Triangulating a hexagon. (Posted on 2010-11-24)
Drawing 9 diagonals within a regular hexagon ensures that all its vertices are connected.

How many triangles there are in the newly created drawing?

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 by my count -- proposed solution | Comment 1 of 7

Let the hexagon be labeled ABCDEF clockwise. Label the intersection nearest side AB and equidistant from A and B be labeled G and the other similarly placed intersections be labeled H, I, J, K and L, clockwise. The midpoint of LG is labeled M, and the similar intersection points labeled clockwise, N, O, P, Q and R. Label the center S.

I've classified the triangles by their size and show only the prototype of a given size in the table below. There are right triangles, as opposed to isosceles or equilateral, and these are shown in alternate (reversed) form and so have 12 occurrences each.

`ABC     6ABD/E  12ACE     2AGB     6ABM/N  12ASB     6ASC     6AMG/L  12ASN/R  12APC/E  12AHD     6      ---       92  total`

I hope I haven't missed any.  I don't think I duplicated any.

 Posted by Charlie on 2010-11-24 13:19:02

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