Symmetry of Any Circle
Corollary: A circle is symmetrical about any of its diameter.
By symmetrical, we mean that the circle can be divided into two congruent parts by any of its diameter. Consider a circle with center O and AB as a diameter:
We need to show that this circle can be divided into two congruent parts by AB.
Proof: Take any point X on the circumference of the circle, and draw XY perpendicular to AB, such that it intersects AB at Z:
In our theorem, we proved that the perpendicular bisector of any chord of a circle passes through its center. Here, we have a chord XY, and a perpendicular AB to XY passing through O. This must mean that AB bisects XY as well, or XZ = YZ.
Thus, for any point X on the circumference of the circle, its mirror image Y in the mirror AB also lies on the circle (reflect on this for a minute). This means that the circle is symmetrical about AB.