Circles

What is a Circle?

A circle is a collection of all points in a flat plane that are at a constant, fixed distance from a fixed point in that same plane. Think of a giant Ferris wheel, a round dinner plate, a coin, or the boundary left behind when you trace a roll of tape with a pencil. The fixed point right in the middle is called the center of the circle.

To master circles, you need to speak the language of geometry. Here are the core terms you need to know:

• Radius: A straight line segment connecting the center of the circle to any point on its outer boundary.
• Chord: A straight line segment whose two endpoints both lie on the circle's outer boundary.
• Diameter: A special chord that passes directly through the center of the circle. It is the longest possible chord in any circle and is exactly twice the length of the radius.
• Secant: A straight line that cuts completely through a circle, intersecting it at exactly two points. Unlike a chord, a secant extends infinitely outside the circle.
• Arc: A continuous portion or piece of the circle's outer curved boundary. A small piece is called a minor arc, while a large piece is called a major arc. Half a circle's boundary is called a semicircle.
• Sector: The region enclosed between two radii and an arc. Think of this as a slice of pizza or cake.
• Segment: The region enclosed between a chord and an arc.

Let us compare the two regional parts of a circle to prevent confusion:

What are the Theorems Governing Chords and Perpendiculars?

When you draw a chord inside a circle and connect its endpoints to either the center or the rim, special angle and distance relationships emerge. These properties are critical for geometric proofs and calculations.

Let us explore the core chord theorems:

1. Angle Subtended by a Chord: If you draw a chord and connect both of its ends to the center of the circle, it forms an angle at the center. The theorem states: Equal chords of a circle subtend equal angles at the center. Conversely, if the angles formed by two chords at the center are equal, the lengths of the chords must be equal.
2. Perpendicular from the Center: If you take a line from the center of the circle and drop it straight down onto a chord at a perfect right angle (90°), a wonderful thing happens: The perpendicular from the center of a circle to a chord bisects the chord. This means it acts as a midpoint divider, splitting the chord into two equal halves.
3. Converse of the Perpendicular Theorem: The straight line drawn from the center of a circle to bisect a chord is automatically perpendicular to the chord.
4. Equal Chords and Distance: The word "distance" in geometry always refers to the shortest, perpendicular path. The theorem states: Equal chords of a circle are equidistant from the center. This means if two different chords have the same length, they sit at the exact same distance away from the center point.

These theorems allow us to form hidden right-angled triangles inside a circle, where we can apply the Pythagorean theorem to find missing lengths.

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.