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John wrote:
This was essentially just another way of looking at Pythagoras's theorem.
In general:
The formula for the area of an equilateral triangle with side $x$ is
$\text{Area} = \frac{x^2\sqrt3}{4}$So with the two triangles with sides a and b respectively, we are looking for a third triangle with area:
$$\frac{c^2\sqrt3}{4} = \frac{a^2\sqrt3}{4} + \frac{b^2\sqrt3}{4} $$