Have you ever thrown a pebble in a pond? Have you ever noticed the formation of ripples in water? Why ripples in water always go in concentric circles and not any other geometrical shape?
This is really interesting to note that when we drop a pebble in a pond, then momentum is transferred to all molecules of water around the stone to adjust the stone. The energy is to spread equally in all directions, hence concentric circles and not squares or a triangles.
1. Equal chords of a circle (or of congruent circles) subtend equal angles at the center.
2. If the angles subtended by the chords of a circle (or congruent circles) at the center (or centers) are equal, then the chords are equal.
3. The perpendicular drawn from the center of the circle to a chord bisects the chord.
4. The line drawn through the center of the circle bisecting the chord is perpendicular to the chord.
5. There is one and only one circle passing through three given non-collinear points.
6. Equal chords of a circle (or of congruent circles) are eqidistant from the center (or centers).
7. Chords equidistant from the center of a circle are equal in length.
8. If two chords of a circle are equal, thten their corresponding arcs are congruent and conversly, if two arcs are congruent, then their corresponding chords are equal.
9. Congruent arcs of a circle subtened equal angles at the center.
10. The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
11. Angles in the same segment of a circle are equal.
12. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, then the four points are concylic.
13. The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.
14. If the sum of a pair of opposite angles of a quadrilateral is 180degrees, the quadrilateral is cyclic.