Hello, Beasticly!
Denis is absolutely correct . . . I'll give it a try.
Note that: \(\displaystyle \,PQ^2\,+\,QR^2\:=\
R^2\;\;\Rightarrow\;\;PQ^2\,+\,QR^2\,-\,PR^2\:=\:0\)
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The areas can be equated like this:
\(\displaystyle [\text{Lunes}]\;=\;\[\Delta PQR]\,+\,[\text{Semicircle on }PQ]\,+\,[\text{Semicircle on }QR]\,- \,[\text{Semicircle on }PR]\)
. . . . . . . . .\(\displaystyle =\;\Delta PQR\,+\,\left[\frac{1}{2}\pi\left(\frac{PQ}{2}\right)^2\right] \,+\,\left[\frac{1}{2}\pi\left(\frac{QR}{2}\right)^2\right] \,-\,\left[\frac{1}{2}\pi\left(\frac{PR}{2}\right)^2\right]\)
. . . . . . . . .\(\displaystyle =\;\Delta PQR\,+\,\frac{\pi}{8}\left(PQ^2\,+\,QR^2\,-\,PR^2)\)
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. . . . . . . . .\(\displaystyle =\;\Delta PQR\)