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High School Mathematics - 2
Touching Circles, Tangents

Theorem: If two Circles touch each other internally or externally, the point of contact and the centers of the circles are collinear.
Data: Two circles with centers A and B touch each other externally at point P (Figure on the left) or internally.
To prove: A, B and P are collinear
Construction: Draw the common tangent RPQ at P. Join AP and BP

For internally touching circles

Theorem: The tangents drawn to a circle from an external point are

  1. Equal
  2. Equally inclined to the line joining the external point and the center
  3. Subtend equal angles at the center
Data: PA and PB are tangents from P to the circle with origin at O
To Prove :
  1. PA=PB
  2. APO= BPO
  3. AOP= BOP

Example: In the figure, XY and PC are common tangents to 2 touching circles. Prove that angle XPY = 90o

Theorem: If a chord(AB) and a tangent(PT) intersect externally, then the product of lengths of the segments of the chord (PA.PB) is equal to the square of the length of the tangent(PT2)from the point of contact(T) to the point of intersection (P).
Given: PT is tangent,AB is chord.
To prove: PA.PB = PT2
Construction: Join O to the mid point M of AB, Join OA.

Directions: Solve the following.