Square
and cross
Proof by rotation. The yellow triangle is rotated 180˚ around the
midpoint M of the side of the square. Convince yourself and justify
each one of the following facts
The
yellow triangle is a right triangle.
Angle α (in the yellow
triangle) and β add to 90˚.
The
length of segment MC is half the length of segment AB.
ABCD
is indeed a square.
There are different ways to convince ourselves that the two pairs of
parallel lines constructed are perpendicular to each other. One such
way
is to use the rotational symmetry of the original figure. Rotating the
square and the dividing segments by 90˚ will result in an identical
figure.
Convince yourself that the little quadrilateral in the middle is indeed
a
square and that it is also congruent to the other four squares.
