- Circle is a closed figure of points which are at a constant distance from a fixed points, centre and constant distance is called radius.
- Two circles are congruent if and only if they have equal radii.
- Equal chords of a circle subtend equal angles at the centre and conversely.
- Two arcs of a circle are congruent if their corresponding chords are equal and conversely.
- The perpendicular drawn from the centre of a circle to a chord bisects the chord.
- The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

**Theorem 1. **There is one and only one circle passing through three non-collinear points.

**Given:-** A, B and C are three non-collinear points

To prove:- One and only one circle are drawn through the points A, B and C.

Construction:- AB and BC are joined PL and QM are drawn perpendicular
bisectors respectively of AB and BC to intersect at O.

OA, OB and OC are joined.

Proof:- O lie on PL, the perpendicular bisector of AB

OA = OB ........................(i)

similarty OB = OC...........................(ii)

from (i) and (ii)

OA = OB = OC = r, say

A circle is drawn taking O as centre and r as radius to pass through A, B and C.

Let there be another centre O’ and another radius s, of circle through A, B and C. O’ must lie on PL and QM. As two lines can intersect at only one point. Hence O and O’ coincide.

OA = O' A = r = s

Hence there is unique circle passing through A, B and C.

Subjects |
Maths (Part-1) by Mr. M. P. Keshari |

Chapter 9 |
Circle |

Chapter 10 |
Tangents to a circle |

Chapter 11 |
Geometrical Construction |

Chapter 12 |
Troigonometry |

Chapter 13 |
Height and Distance |

Chapter 14 |
Mensuration |

Chapter 15 |
Statistics |

Chapter 16 |
Probability |

Chapter 17 |
Co-ordinate Geometry |