Continuing today's earlier remarks . . .
One approach to the mystery —
IF one could inscribe in a semicircle, upon the diameter of the circle,
a right triangle whose hypotenuse is the diameter of the circle and
whose area is exactly half of the semicircle's area
THEN clearly one could do the same on the diametrically opposite side
of the circle and form a rectangle whose area is half that of the circle . . .
AND then convert that rectangle to a square, as below . . .
. . . and finally , as in the first geometric problem in the Meno , one
could use the new square (green in the figure above) to easily construct
a square with double the area.
That square — from the matrix of "Plato's diamond" —
would thus have the same area as the circle.
Thus, granted the hypothesis that the first triangle pictured
above has half the area of the semicircle in which it is inscribed . . .
One would have achieved the seemingly impossible, and squared the circle.