Circles Through Three Points and Cyclic Quadrilaterals

What are the Laws for Three Points and Cyclic Quadrilaterals?

Now we look at how circles interact with multiple points and multi-sided shapes like quadrilaterals. Two beautiful geometric truths emerge here:

1. Circle Through Three Points: If you pick any two distinct points in space, you can draw infinite different circles through them. But what if you have three points?

• If the three points lie on a perfectly straight line (collinear points), you can never draw a circle that passes through all of them.
• If the three points do not lie on a straight line (non-collinear points), the theorem states: There is one and only one circle passing through three given non-collinear points. This unique circle can be found by constructing the perpendicular bisectors of the lines joining the points; their crossing point is the center.

1. Cyclic Quadrilaterals: A quadrilateral is a four-sided shape. A cyclic quadrilateral is a special quadrilateral where all four of its vertices (corners) sit perfectly on the curved outer boundary of a single circle.

• The defining theorem states: The sum of either pair of opposite angles of a cyclic quadrilateral is always 180°. This means they are supplementary.
• Conversely, if the opposite angles of a four-sided shape add up to 180°, the shape is guaranteed to be a cyclic quadrilateral.