Spike in the Hive

Our next Catriona Shearer puzzle.
12.
Spike in the Hive


Two of the regular hexagons are identical; the third has area 10. What’s the area of the red triangle?

Spoiler!

First, what’s the area of the big hexagons?

There’s a little equilateral triangle in the middle between the hexagons, and its equilateralness means a big hex side is equal to a little hex side plus a triangle side (H=h+t, from the upper big hex), but also a big hex side is two little hex sides minus a triangle side (H=2h-t, from the lower big hex), and that means t = h/2 and H = 3/2 h. So the area of a big hex is 9/4\times 10.

Now, the red triangle area: It’s half base times height, so it’s twice the area of a triangle with the same base and half the height, or twice the area of the same triangle sheared parallel to its base:

And that last triangle clearly is 1/3 the area of the hexagon it’s inscribed in. So A = 2/3\times 9/4\times 10 = 15.

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