A point P is positioned inside regular hexagon ABCDEF so that CP < AP. Triangles BPC, BPE, and APE have areas 7, 12, and 28, respectively. Find the area of the hexagon.
Some poking around numerically (trial and error) gave:
P = (-2.693934 -0.518054) from the (0,0) center on a hexagon
of side = 5.78666 to produce the areas specified. The full hexagon
has area = 86.99776. All of this is with about 0.001 uncertainty.
If there is a construction method to prove this, I didn't find it.
I wonder if the author may have intended a simpler problem with areas of
7, 14, 28. These would give a set of area ratios, 1:2:4, which could have
been nicely supplied using P'=(-0.5,0) on, say, a unit side hexagon.
P and P' are compared on a unit hexagon here:
https://www.desmos.com/calculator/6mn5c3z2zm
In this alternate case, to get the specific areas (7, 14, and 28),
we would use a side of 56/sqrt(3) yielding a total hexagonal area of
84 (= 28 * 3)
Edited on December 27, 2024, 8:57 pm
soln
