Convex hexagon ABCDEF is equiangular but has no two sides the same length. Its sides in some order are 1, 2, 3, 4, 5 and 6 units long. If AB=1 and CD>BC, what are the lengths of BC, CD, DE, EF and FA?

Another convex hexagon is also equiangular and has sided measuring 1, X, 3, 4, 5, and 6 units long in that order going clockwise. What is the measure of X?

In an equiangular hexagon there are 3 pairs of parallel sides.

Given a pair of parallel sides, the consecutive pairs on either side must have the same sum.   This results in the following:

AB + BC = DE + EF

BC + CD = EF + FA

CD + DE = FA + AB

To solve part 2 the first equation becomes

1+x=4+5 so x=8

To solve part 1 it seemed reasonable to make opposite sides of similar length.  The order of the sides that work are

1, 4, 5, 2, 3, 6

another hexagon which doesn't satisfy CD>BC is

1, 5, 3, 4, 2, 6

-Jer

Posted by Jer on 2004-10-15 16:30:33