The triangular insets fitted into their metal plates.
B. The Theorem of Pythagoras: In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the two sides.
Material: The material illustrates three different cases:
First case: In which the two sides of the triangle are equal.
Second case: In which the two sides are in the proportion of 3:4.
Third case: General.
First case: The demonstration of this first case affords an impressive induction.
In the frame for this, shown below, the squares of the two sides are divided in half by a diagonal line so as to form two triangles and the square of the hypotenuse is divided by two diagonal lines into four triangles. The eight resulting triangles are all identical; hence the triangles of the squares of the two sides will fill the square of the hypotenuse; and, vice versa, the four triangles of the square of the hypotenuse may be used to fill the two squares of the sides. The substitution of these different pieces is very interesting, and all the more because the triangles of the squares of the sides are all of the same color, whereas the triangles formed in the square of the hypotenuse are of a different color.