I’m Up and Down, and Round and Round

1. Chapter at a glance

• A circle is the set of all points in a plane equidistant from a fixed point (the centre); the common distance is the radius.
• A circle possesses complete rotational symmetry about its centre and reflection symmetry across every diameter.
• Infinitely many circles pass through two given points; their centres lie on the perpendicular bisector of the segment joining the points.
• Exactly one circle (the circumcircle) passes through three non-collinear points; its centre (circumcentre) is the intersection of the perpendicular bisectors of the sides of the triangle formed by the points.
• Equal chords subtend equal angles at the centre (and conversely); the perpendicular from the centre to a chord bisects the chord (and conversely).
• Chords of equal length are equidistant from the centre (and conversely); the longer of two unequal chords lies closer to the centre.
• The angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the remaining part of the circle; the angle in a semicircle is 90°.
• Four points are concyclic if and only if a line segment joining two of them subtends equal angles at the other two points on the same side; a cyclic quadrilateral has opposite angles summing to 180° (and conversely).

2. Definitions, theorems and results

Definitions (exactly as framed)
- A circle is the set of all points on the plane that are equidistant from a given point on that plane (the centre).
- The distance from the centre to any point on the circle is the radius.
- A chord is a line segment joining two points on the circle. A chord passing through the centre is a diameter.
- An arc is a connected portion of the circle defined by two endpoints and the curve along the circle. The larger arc is the major arc; the smaller is the minor arc.
- Points lying on the same circle are concyclic. A quadrilateral whose vertices are concyclic is a cyclic quadrilateral.
- The circumcentre of △ABC is the centre of the unique circle passing through A, B and C; the circle is the circumcircle.

Theorems (stated precisely; all require proof in the exam unless noted)
- Theorem 1: There is a unique circle passing through three non-collinear points.
- Theorem 2: Equal chords of a circle subtend equal angles at the centre.
- Theorem 3: Chords of a circle that subtend equal angles at the centre are equal.
- Theorem 4: The line joining the centre of a circle and the midpoint of a chord is perpendicular to the chord.
- Theorem 5: The perpendicular from the centre of a circle to a chord bisects the chord.
- Theorem 6: Chords of a circle having the same length are all at the same distance from the centre.
- Theorem 7: Chords of a circle that are equidistant from the centre have equal length.
- Theorem 8: If AB > DE are two chords of a circle with centre C, then the perpendicular distance from C to AB is less than the perpendicular distance from C to DE.
- Theorem 9: The angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the circle outside the arc.
- Corollary (from Th. 9): The angle subtended by a diameter at any point on the circle is 90°.
- Theorem 10: If a line segment AB subtends equal angles at two other points C and D on the same side of AB, then A, B, C, D are concyclic.
- Theorem 11: The sum of each pair of opposite angles of a cyclic quadrilateral is 180°.
- Theorem 12: If two opposite angles of a quadrilateral sum to 180°, then the quadrilateral is cyclic.

3. Formula sheet

4. Solved-example patterns

• Constructing the circumcircle of △ABC
  Steps: (i) Draw the triangle. (ii) Construct perpendicular bisectors of at least two sides. (iii) Their intersection is the circumcentre O. (iv) Draw circle with radius OA. (v) State whether O lies inside/outside/on the hypotenuse according to the type of triangle.

• Proving two chords equal or angles equal using congruence
  Steps: (i) Draw radii to chord endpoints. (ii) Identify SSS/SAS/RHS congruence (equal radii + given equal chords/angles). (iii) Conclude corresponding parts equal.

• Finding chord length or distance from centre
  Steps: (i) Draw radii to chord ends and perpendicular from centre to chord (bisects chord). (ii) Apply Pythagoras in the right triangle formed. (iii) Solve for unknown length/distance.

• Finding angles using the “angle at centre = twice angle at circumference” or cyclic-quadrilateral property
  Steps: (i) Identify the arc and the relevant inscribed angle or opposite angles. (ii) Apply the double-angle relation or sum-to-180° relation. (iii) Use linear-pair or exterior-angle facts when needed.

• Proving four points concyclic or a quadrilateral cyclic
  Steps: (i) Show equal angles subtended by the same segment or opposite angles sum to 180°. (ii) Invoke Theorem 10 or 12. (iii) Rule out inside/outside positions by contradiction if required.

5. Common mistakes and exam pitfalls

• Forgetting that three collinear points do not determine a circle (or that the perpendicular bisectors become parallel).
• Assuming the circumcentre is always inside the triangle (must check acute/obtuse/right case).
• Missing the “same side of the chord” condition when applying the concyclic test or angle-in-same-segment property.
• Confusing major/minor arc when stating the central angle or inscribed angle.
• Omitting the radius when applying Pythagoras to find chord length (or forgetting to double the half-chord).
• Writing “opposite angles sum to 180°” without verifying both pairs or without stating the cyclic-quadrilateral converse correctly.
• Neglecting units (cm, °) or sign/direction when comparing distances of chords from the centre.