Class 9 — Chapter 5: I’m Up and Down, and Round and Round

5.1 Definitions & Basic Concepts

Circle: The set (or locus) of all points on a two-dimensional plane that are equidistant from a given point (the centre).
Radius: The constant distance from the centre to any point on the circle.
Chord: A line segment joining any two points on the circle.
Diameter: A chord passing through the centre of the circle. It is the longest chord of the circle and is equal to twice the radius.

5.2 Symmetries of a Circle

Rotational Symmetry: A circle has complete rotational symmetry about its centre; it looks identical when rotated by any angle.
Reflection Symmetry: Any line passing through the centre (i.e., any diameter) acts as a line of reflection symmetry.

5.3 Number of Circles Through Given Points

Through Two Points: Infinitely many circles can pass through two distinct points $A$ and $B$. Their centres all lie on the perpendicular bisector of segment $AB$.
Through Three Points (Theorem 1): There is a unique circle passing through three non-collinear points. No circle can pass through three collinear points.
Circumcircle: The unique circle passing through all three vertices of a triangle.
Circumcentre: The centre of the circumcircle, located at the intersection of the perpendicular bisectors of the triangle's sides.
Circumcentre Location:
- Acute-angled triangle: Inside the triangle.
- Obtuse-angled triangle: Outside the triangle.
- Right-angled triangle: At the midpoint of the hypotenuse.

5.4 Chords and Angles Subtended

Theorem 2: Equal chords of a circle subtend equal angles at the centre of the circle.
Theorem 3 (Converse): Chords of a circle that subtend equal angles at the centre are equal in length.

5.5 Midpoints and Perpendicular Bisectors of Chords

Theorem 4: The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
Theorem 5 (Converse): The perpendicular from the centre of a circle to a chord bisects the chord.

5.6 Distance of Chords from the Centre

Theorem 6: Chords of a circle having the same length are equidistant from the centre.
Theorem 7 (Converse): Chords of a circle that are equidistant from the centre are equal in length.
Theorem 8: Out of two unequal chords, the longer chord is closer to the centre (i.e., if $AB > DE$, then the perpendicular distance to $AB$ is less than the perpendicular distance to $DE$).

5.7 Angles Subtended by an Arc

Arc: A connected portion of a circle. Divided into minor arc (smaller) and major arc (larger).
Theorem 9: The angle subtended by an arc at the centre of the circle is double the angle subtended by the same arc at any point on the remaining part of the circle.
Angles in the Same Segment: Angles subtended by an arc at different points on the remaining part of the circle are equal ($ \angle AEB = \angle ADB = \angle AFB = \frac{1}{2}\angle ACB$).
Semicircle Angle Corollary: The angle subtended by a diameter at any point on the circle is a right angle ($90^\circ$).

5.8 Concyclicity & Cyclic Quadrilaterals

Concyclic Points: A set of points that lie on the same circle.
Theorem 10: If a line segment joining two points subtends equal angles at two other points on the same side of the segment, then all four points are concyclic.
Cyclic Quadrilateral: A quadrilateral whose vertices all lie on a circle.
Theorem 11: The sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$.
Theorem 12 (Converse): If the sum of a pair of opposite angles of a quadrilateral is $180^\circ$, then the quadrilateral is cyclic.

Mathematics: Class 9 — Chapter 5: I’m Up and Down, and Round and Round