The figure shows a square ABCD
inscribed in a circle O. S is the area of the right triangle
AOB, S1 is the area of the lune of Hippocrates
bounded by the semi-circle of diameter AB and the arc AB. A
radius OE extended cuts semi-circle AB at F. S2 is
the area of the
kite EHFG (H on OA extended, G on OB extended). S3
and S4 are the areas of the triangles EFH and EFG,
respectively. S5 and S6 are the areas of
the curved triangles AFE and BFE.
Prove that (1) S3
= S5; (2) S4 = S6; (3) S = S1
= S2; (4) AB2 = 4.AH.BG.
contributed by Markus Heisss, Wurzburg, Bavaria. Published in:
"Die Wurzel - Zeitschrift fur Mathematik, Heft 11/2015",