In the figure below, given a right triangle ABC
of area S, BC = a, AC = b, AB = c. S_{1}, S_{2}, S_{3} are the areas of
the shaded squares built on the catheti (legs) b, c and the hypotenuse a,
respectively. S_{4}, S_{5}, S_{6} are the areas of the shaded squares.
S_{7}, S_{8}, S_{9} are the areas of the shaded quadrilaterals, S_{10}, S_{11},
S_{12} are the areas of the shaded squares built on the lines FG,
HM, DE, respectively. S', S'', S''', K are the areas of the
shaded triangles. Prove the 15 conclusions below:
Reference: Loomis, Elisha Scott. The Pythagorean Proposition,
Classics in Mathematics Education Series. National Council of Teachers of Mathematics, 1968.
Elisha Scott Loomis, Ph.D., LL.B., was professor of mathematics at Baldwin University for the period 188595 and head of the mathematics department at West High School, Cleveland, Ohio, for the period 18951923.
