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### High School Mathematics - 28.9 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 Equal Equally inclined to the line joining the external point and the center Subtend equal angles at the center Data: PA and PB are tangents from P to the circle with origin at O To Prove : PA=PB APO= BPO AOP= BOP Proof: 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.
 Q 1: Answer: Q 2: Answer: Q 3: A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AD+BC = AB+CDAnswer: Q 4: Prove that the line segment joining the points of contact of two parallel tangents to a circle is diameter of the circle.Answer: Q 5: Answer: Q 6: Answer: Q 7: Answer: Q 8: Tangents PQ and PR are drawn to the circle from an external point P. If PQR = 60 degrees prove that the length of the chord QR = length of the tangentAnswer: Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!