Circle Part 01

 Circles passing through one, two, three points : 

  1. Infinite circles pass through one point.
  2. Infinite circles pass through two distinct points.
  3. There is a unique circle passing through three non-collinear points.
  4. No circle can pass through 3 collinear points.
Tangent theorem : 
  • A tangent at any point of a circle is perpendicular to the radius at the point of contact.
Converse of tangent theorem :
  • A line perpendicular to a radius at its point on the circle is a tangent to the circle.
Tangent segment theorem : 
  • Tangent segments drawn from an external point to a circle are congruent.
Touching circles :
  • If two circles in the same plane intersect with a line in the plain in only one point, they are said to be touching circles.
  • The line is their common tangent.
  • The point common to the circles and the line is called their common point of contact.
Theorem of touching circles :
  • If two circles touch each other, their point of contact lies on the line joining their centres.
Remember this!
  1.  The point of contact of the touching circles lies on the line joining their centres.
  2. If the circles touch each other externally, distance between their centres is equal to the sum of their radii.
  3. The distance between the centres of the circles touching internally is equal to the difference of their radii.
Arc of a circle : 
  • A secant divides a circle in two parts.
  • Any one of these two parts and the common points of the circle and the secant constitute an arc of the circle.
  • If the centre of a circle is on one side of the secant then the arc on the side of the centre is called ‘major arc’.
  • The arc which is on the other side of the centre is called ‘minor arc’.
Central angle : 
  • When the vertex of an angle is the centre of a circle, it is called a central angle.
Measure of an arc : 
  1. Measure of a minor arc is equal to the measure of its corresponding central angle.
  2. Measure of major arc = 360° - measure of corresponding minor arc.
  3. Measure of a semi circular arc, that is of a semi circle is 180°.
  4. Measure of a complete circle is 360°.
Congruence of arcs :
  • When two coplanar figures coincide with each other, they are called congruent figures.
  • Two arcs are congruent if their measures and radii are equal.
  • When two arcs are of the same radius and same measure, they are congruent.
Property of sum of measures of arcs : 
  • Theorem: The chords corresponding to congruent arcs of a circle ( or congruent circles) are congruent.
  • Theorem: Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent.
Inscribed angle : 
  • An inscribed angle is the an gle formed in the interior of a circle when two chords intersect on the circle.
Inscribed angle theorem :
  • The measure of an inscribed angle is half of the measure of the arc intercepted by it.
  • The measure of an angle subtended by an arc at a point on the circle is half of the measure of the angle subtended by the arc at the centre.
Corollaries of inscribed angle theorem :
  1. Angles inscribed in the same arc are congruent.
  2. Angle inscribed in a semicircle is a right angle.
Cyclic quadrilateral : 
  • If all vertices of a quadrilateral lie on the same circle then it is called a cyclic quadrilateral.
Theorem of cyclic quadrilateral : 
  • Opposite angles of a cyclic quadrilateral are supplementry.
Corollary of cyclic quadrilateral theorem : 
  • An exterior angle of a cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle.
Converse of cyclic quadrilateral theorem : 
  • Theorem : If a pair of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic.
+ Points : 
  • For every triangle there exists a circumcircle but there may not be a circumcircle for every quadrilateral.
  • Theorem : If two points on a given line subtend equal angles at two distinct points which lie on the same side of the line, then the four points are concyclic.
  • If two chords of a circle intersect each other in the interior of a circle then the measure of the angle between them is half the sum of measures of arcs intercepted by the angle and its opposite angle.
  • if two lines containing chords of a circle intersect each other outside the circle, then the measure of angle between them is half the difference in measures of the arcs intercepted by the angle.
  • Any rectangle is a cyclic quadrilateral.

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