Comprehensive theory, key formulas, diagrams, and memory aids for Circles.
A circle is perhaps the most perfect and symmetrical shape in geometry. From wheels and coins to the orbits of planets (approximately), circles are everywhere. In this chapter, we will study the mathematical properties of circles, focusing on chords, arcs, and the angles they subtend.
A circle is the collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre) in the plane.
The chords of a circle possess several fascinating geometric properties, particularly in relation to the centre of the circle.
Theorem 1: Equal chords of a circle subtend equal angles at the centre. Proof Concept: If $AB$ and $CD$ are two equal chords of a circle with centre $O$. In $\triangle AOB$ and $\triangle COD$: $OA = OC$ (Radii) $OB = OD$ (Radii) $AB = CD$ (Given) Therefore, $\triangle AOB \cong \triangle COD$ (by SSS congruence). Thus, $\angle AOB = \angle COD$.
Theorem 2 (Converse): If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
Theorem 3: The perpendicular from the centre of a circle to a chord bisects the chord. If $OM \perp AB$ (where $AB$ is a chord), then $AM = MB$. Proof Concept: Draw radii $OA$ and $OB$. In right triangles $OMA$ and $OMB$, $OA = OB$ (hypotenuse) and $OM = OM$ (common side). By RHS congruence, $\triangle OMA \cong \triangle OMB$. Therefore, $AM = MB$.
Theorem 4 (Converse): The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Theorem 5: Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres). Theorem 6 (Converse): Chords equidistant from the centre of a circle are equal in length.
How many circles can pass through a given point? Infinitely many. How many through two given points? Still infinitely many. But what about three points?
Theorem: There is one and only one circle passing through three given non-collinear points. Proof Concept: Let $A, B, C$ be three non-collinear points. Draw perpendicular bisectors of line segments $AB$ and $BC$. Since $A, B, C$ are non-collinear, these perpendicular bisectors will intersect at exactly one point, say $O$. Since $O$ lies on the perpendicular bisector of $AB$, $OA = OB$. Since $O$ lies on the perpendicular bisector of $BC$, $OB = OC$. Therefore, $OA = OB = OC$. If we draw a circle with centre $O$ and radius $OA$, it must pass through $B$ and $C$. Because two lines can intersect at only one point, $O$ is unique, and thus the circle is unique.
Note: If three points are collinear, a circle cannot pass through all three of them.
An arc of a circle subtends an angle at the centre and also at any point on the remaining part of the circle.
Theorem 1: The angle subtended by an arc at the centre is double the angle subtended by it at any remaining part of the circle. Proof Concept: Let an arc $PQ$ subtend $\angle POQ$ at the centre and $\angle PAQ$ at a point $A$ on the remaining circle. By joining $A$ to $O$ and extending it to $B$, we form triangles. Using the exterior angle theorem of a triangle (exterior angle is sum of interior opposite angles), and knowing that radii $OA=OP$ and $OA=OQ$ make the triangles isosceles, we can prove that $\angle POQ = 2 \angle PAQ$.
Theorem 2: Angles in the same segment of a circle are equal. Since the angle at the centre is fixed for a given arc, any angle formed on the remaining part of the circle by that arc must be exactly half of the central angle. Because they are all half of the same central angle, they must all be equal to each other.
Theorem 3: Angle in a semicircle is a right angle ($90^\circ$). The angle subtended by a diameter at the centre is $180^\circ$ (a straight line). Therefore, the angle subtended by the diameter at any point on the circle is $180^\circ / 2 = 90^\circ$.
A quadrilateral is called cyclic if all its four vertices lie on a circle.
Theorem 1: The sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$. (They are supplementary). If $ABCD$ is a cyclic quadrilateral, then $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.
Theorem 2 (Converse): If the sum of a pair of opposite angles of a quadrilateral is $180^\circ$, the quadrilateral is cyclic.
The circle is a shape defined by a single distance rule (radius), yet it gives rise to a vast array of interconnected theorems. The properties of chords, the constant nature of angles in the same segment, and the supplementary nature of cyclic quadrilaterals form a comprehensive toolkit for advanced geometric analysis.