Drawing 9 diagonals within a regular hexagon ensures that all its vertices are connected.
How many triangles there are in the newly created drawing?
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 6
ABD/E 12
ACE 2
AGB 6
ABM/N 12
ASB 6
ASC 6
AMG/L 12
ASN/R 12
APC/E 12
AHD 6

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

Posted by Charlie
on 20101124 13:19:02 