The circle is the most symmetric of all geometric shapes — the locus of all points at a fixed distance from a centre. It appears in nature more than any other curve, and its properties have fascinated mathematicians since antiquity. This chapter covers the full geometry of the circle: arc and chord theorems, tangents, secants, and several advanced concepts including inverse points, the radical axis, and the Apollonius circle.
Key Takeaways
- The perpendicular bisector of any chord always passes through the centre of the circle.
- An inscribed angle is measured by half the intercepted arc; an angle inscribed in a semicircle is always a right angle.
- The tangent to a circle at any point is perpendicular to the radius drawn to that point.
- Two tangents drawn from an external point to a circle are equal in length.
- If two chords intersect inside a circle, the products of their segments are equal.
What is a Circle?
The circle is a fundamental geometric shape — the most aesthetic of all curves and the most symmetric. Its construction using a compass is granted by Euclid's third postulate. Of all plane shapes with a fixed perimeter, the circle bounds the greatest area. Its geometric properties have been a source of curiosity and pleasure for mathematicians throughout history.
Definitions
Circle — a closed curve lying in a plane, all of whose points are equally distant from a fixed point called the centre. Equivalently, the locus of a point in a plane at a given distance from a fixed point.
Diameter — a straight line through the centre, terminated at each end by the circle. All diameters of the same circle are equal, each equalling two radii.
Radius — a straight line from the centre to the circumference. All radii of the same circle are equal.
Circumference — the distance around the perimeter of the circle. If the diameter is 1, the circumference is π.
Arc — any portion of a circle. An arc equal to half the circle is a semicircle; less than a semicircle is a minor arc; greater than a semicircle is a major arc.
Central Angle — an angle with its vertex at the centre and its sides as radii. A central angle intercepts the arc cut off by its sides.
Circular Arc Theorems
- In the same circle or equal circles, equal central angles intercept equal arcs; of two unequal central angles, the greater intercepts the greater arc.
- In the same circle or equal circles, equal arcs subtend equal central angles; of two unequal arcs, the greater subtends the greater central angle.
Chord
A chord is a straight line with both extremities on the circle, subtending the arcs it cuts from the circle.
Chord Theorems: - Equal arcs are subtended by equal chords; of two unequal arcs, the greater is subtended by the greater chord. - Equal chords subtend equal arcs; of two unequal chords, the greater subtends the greater arc. - A line through the centre perpendicular to a chord bisects the chord and both arcs subtended by it. - Corollary: A diameter bisects the circle. - Corollary: The perpendicular bisector of a chord passes through the centre and bisects both arcs. - Corollary: Equal chords are equidistant from the centre; chords equidistant from the centre are equal. - Corollary: The greater of two unequal chords is at the lesser distance from the centre. - Corollary: A diameter is greater than any other chord.
Secant and Tangent
Secant — a straight line intersecting a circle at two points.
Tangent — a straight line of unlimited length having exactly one point in common with a circle.
Theorems: - A line perpendicular to a radius at its extremity on the circle is tangent to the circle. - Corollary: A tangent is perpendicular to the radius drawn to the point of contact. - Corollary: A perpendicular to a tangent at the point of contact passes through the centre.
Concentric Circles
Two circles with the same centre are concentric.
Theorems: - Two parallel lines intercept equal arcs on a circle. - Through three points not in a straight line, one and only one circle can be drawn. - The tangents to a circle drawn from an external point are equal and make equal angles with the line joining the point to the centre. - Corollary: Two circles can intersect in at most two points.
Tangent Circles
Line of Centres — the line determined by the centres of two circles.
Tangent Circles — two circles both tangent to the same line at the same point. Circles are internally tangent if they lie on the same side of the tangent line, and externally tangent if on opposite sides. If two circles are tangent to each other, the line of centres passes through the point of contact.
Angle Theorems: - In the same circle or equal circles, two central angles have the same ratio as their intercepted arcs. - An inscribed angle is measured by half the intercepted arc. - An angle formed by two chords intersecting within the circle is measured by half the sum of the intercepted arcs. - An angle formed by a tangent and a chord drawn from the point of contact is measured by half the intercepted arc. - An angle formed by two secants, a secant and a tangent, or two tangents from an external point is measured by half the difference of the intercepted arcs. - Corollary: An angle inscribed in a semicircle is a right angle. - Corollary: Angles inscribed in the same or equal segments are equal. - Corollary: If a quadrilateral is inscribed in a circle, opposite angles are supplementary.
Euclidean Constructions
Finding the Centre of a Circle
Draw any chord AB and bisect it at D. Erect a perpendicular at D and extend it to meet the circle at E. Bisect CE at F. The point F is the centre. (Proof: any point not equidistant from A and B cannot be the centre, and the perpendicular bisector of any chord passes through the centre.)
Drawing a Tangent from an External Point
Given external point A and circle with centre E: join AE, construct a circle with diameter AE, mark its intersections with the original circle — the line from A through either intersection is the required tangent. The construction relies on the fact that an angle inscribed in a semicircle is a right angle.
Completing a Circle from a Segment
Given arc ABC, bisect the chord AC at D, erect a perpendicular, and use the intersection of this perpendicular with the perpendicular bisector of any other chord to locate the centre. The full circle is then described with that centre and radius.
Advanced Geometry
Inverse Points
Two points collinear with the centre of a circle whose distances from the centre multiply to give the square of the radius are inverse points. One lies inside and one outside the circle. A point on the circle is its own inverse.
Properties: - Two inverse points divide the corresponding diameter harmonically. - The ratio of the distances of any point on the circle from two given inverse points is constant.
Orthogonal Circles
Two circles are orthogonal if the square of the distance between their centres equals the sum of the squares of their radii.
Properties: - In two orthogonal circles, the radii through a common point are perpendicular. - If two circles are orthogonal, the radius of one through a common point is tangent to the other. - If two circles are orthogonal, any two points of one collinear with the centre of the other are inverse points for that other circle.
Circle of Apollonius
Apollonius of Perga showed that a circle can be defined as the locus of points in a plane having a constant ratio (other than 1) of distances to two fixed points A and B. For every such ratio r > 0 there is a different Apollonian Circle. This gives an alternative definition of the circle alongside the classical one.
Poles and Polars
Given a fixed point P and a circle, draw any chord through P. The two endpoints of the chord and P itself determine a fourth point — the harmonic conjugate of P with respect to those two endpoints. As the chord rotates around P, this fourth point traces a straight line: the polar of P with respect to the circle. The fixed point P is called the pole of that line. When P lies outside the circle, the polar is the chord of contact — the line joining the two points where the tangents from P touch the circle.
Power of a Point
The product of the signed distances from a given point to any two points on the circle that are collinear with it is a constant — the power of the point with respect to the circle.
Radical Axis
The locus of a point whose powers with respect to two given circles are equal is a straight line perpendicular to the line of centres — the radical axis of the two circles.
Coaxial Circles
A group of circles forming a coaxial pencil share the same radical axis for any two circles in the group.
Frequently Asked Questions
What is the difference between a chord, a secant, and a tangent?
A **chord** is a line segment whose both endpoints lie on the circle — it lives entirely within or on the circle. A **secant** is an infinite line (or ray) that crosses the circle at two distinct points — it extends beyond the circle. A **tangent** is an infinite line that touches the circle at exactly one point and does not cross it; at that point it is perpendicular to the radius.Why is an angle inscribed in a semicircle always a right angle?
An inscribed angle equals half the central angle subtending the same arc. A semicircle subtends a central angle of 180° (a straight line). Half of 180° is 90°. Therefore any angle inscribed in a semicircle is always exactly 90°. This is sometimes called Thales' theorem.What is the power of a point?
For a fixed point P and a circle with centre O and radius r, the power of P equals OP² − r². If P is outside the circle, the power is positive and equals the square of the tangent length from P to the circle. If P is inside, the power is negative. If two chords (or secants) pass through P, the products of their signed segment lengths are equal — both equal the power of P. This is a unifying theorem that connects chords, secants, and tangents.How do you find the equation of a circle?
A circle with centre (a, b) and radius r is described by (x − a)² + (y − b)² = r². Expanding gives x² + y² − 2ax − 2by + (a² + b² − r²) = 0, the general form. Given three points on the circle, substituting each gives three equations in three unknowns (a, b, r²), which can always be solved uniquely — confirming that exactly one circle passes through any three non-collinear points.What are orthogonal circles?
Two circles are **orthogonal** if they meet at right angles at each of their intersection points — meaning the radius of one circle at the point of intersection is tangent to the other circle. This happens when the square of the distance between their centres equals the sum of the squares of their radii: d² = r₁² + r₂². Orthogonal circles have a beautiful relationship: the two intersection points of one circle are inverse points with respect to the other.Conclusion
The circle is the most symmetric figure in Euclidean geometry, and its theorems — from the inscribed angle theorem to the power of a point — follow with remarkable elegance from a single definition. Every chord, tangent, and secant relationship reduces to a ratio or a product, reflecting the circle's role as the boundary between the algebraic world of ratios and the metric world of lengths. These theorems reappear throughout the guide, from trigonometry and conic sections to the study of the sphere.
The next chapter examines the geometry of proportional figures — Similarity and Proportion.