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Intermediate

Below the tree line

A triangle is split in two. What is the area of the green quadrilateral in terms of lengths a and b?

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Solution

The green area is (b2-a2)/(4√3).

The trick is to note that the bottom angles add up to 60°. We can therefore fit together three copies of the quadrilateral as shown to form two concentric equilateral triangles of sides a and b.

Now it is simply a matter of subtracting the small triangle area a2√3/4 from the large b2√3/4 and dividing by 3.

Poem

Below the tree line
Some disturbed vegetation
Of trees in a strange distribution
To measure direct interaction
With the rain and below average precipitation
It’s the highest elevation
Especially to avoid a gradual extinction
The tree line is the natural solution

Bella

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