Circle Geometry

Circle Geometry

1. Angles

   • • Equal angles at the centre of the circle stand on equal arc.
     • Don't need to prove
     • IMPORTANT TO REMEMBER
   • • Theorem- The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.
     • Don't need to prove
     • IMPORTANT TO REMEMBER
   • • Theorem- The angle in a semicircle is a right angle.
     • IMPORTANT TO REMEMBER
     • Need to prove
   • • Need to prove
     • IMPORTANT TO REMEMBER
     • shorter as, Angles in the same segment are equal.

2. Chords

   • • Need to prove
     • Theorem- The perpendicular bisector of a chord passes through the centre of the circle.
   • • need to prove
     • Theorem 1- The perpendicular from the centre of a circle to a chord bisects the chord.
     • Theorem 2- The line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.
     • Need to prove
   • • Need to prove
     • Theorem- Chords that are equidistant from the centre of the circle are equal.
       • with this question i saw the conce

3. Cyclic Quadrilaterals

   • • Theorem- The opposite angles of a cyclic quadrilateral are supplementary.
     • Need to prove
     • IMPORTANT TO REMEMBER
   • • Theorem- An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
     • Need to prove
   • • Theorem- If the opposite angles in a quadrilateral are supplementary then the quadrilateral is cyclic.
     • This is also a test for four points to be concyclic.
     • Need to prove
   • • Theorem- If two points lie on the same side of an interval, and the angles subtended at these points by the interval are equal, then the two points and the endpoints of the interval are concyclic.
     • Need to prove
   • • Theorem- Any three non-collinear points lie on a unique circle, whose centre is the point of concurrency of the perpendicular bisectors of the intervals joining the points.
     • Need to prove

4. Tangents

   • • Theorem- Two circles touch if they have a common tangent at the point of contact.
     • Need to prove
   • • Theorem- The tangent to a circle is perpendicular to the radius drawn to the point of contact.
     • Don't need to prove
   • • Theorem- Tangents to a circle from an external point are equal.
     • Need to prove
     • Looks like Birds beak
     • Need to prove
   • • Don't need to prove
     • Theorem- When two circles touch, their centres and the point of contact are collinear.
   • • Theorem- Angle in the alternate segment.
     • The sail boat
     • Need to prove
     • MOST IMPORTANT

5. Intersecting Secants and Chords

   • • Theorem- The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point.
     • Need to prove
   • • Need to prove
     • Theorem- The products of the intercepts of two intersecting secants to a circle from an external point are equal.
   • • Need to prove
     • Theorem- The products of the intercepts of two intersecting chords are equal.

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