Q. 1. The diagonals of a parallelogram ABCD intersect in a point E. show that the circumcircles
touch each other at E.
Q. 2. If a line touches a circle and from the point of contact a chord is drawn, the angles which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segment.
Q. 3. The radius of the incircle of a triangle is 4cm and the segments into which one side is divided by the point of contact are 6cm. and 8cm, determine the other two sides of the triangle.
Q. 4. If PAB is a secant to a circle intersecting it at A and B and PT is a tangent. Then prove that PA.PB= PT2
Q. 5. Given two concentric circles of radii a and b where a>b. find the length of a chord of larger circle which touches the other.
Q. 6. If a line is drawn through an end point of a chord of a circle so that the angle formed by it with the chord is equal to the angle subtended by the chord in the alternate segment then the line is a tangent to the circle.
7. Two circles intersect each other at two points A and B. at A, tangents AP and AQ to the two circles are drawn which intersect other circles at the points P and Q respectively. Prove that AB is the bisector of angle PBQ.
Q. 8. If two chords of a circle intersect inside or outside the circle, then the rectangle formed by the two parts of one chord is equal to in area to the rectangle formed by the two parts of the other
Q. 9. Given a right triangle ABC, a circle is drawn with diameter AB intersecting hypotenuse AC at the point P. show that the tangent to the circle at P bisects the side BC.
Q. 10. Two rays ABP and ACQ are intersected by two parallel lines in B, C and P, Q respectively. Prove that the circumcircles of 
touch each other at A.
Q. 11. A circle touches all the four sides of a quadrilateral ABCD. Prove that the angles subtended at the centre of the circle by the opposite sides are supplementary.
Q. 12. Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.
Q. 13. Two circles with radii a and b touch each other externally. Let c be the radius of a circle which touches these two circles as well as a common tangent to two circles. Prove that 
Q. 14. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
Q. 15. let A be one point of intersection of two intersecting circles with centres O and Q. the tangent at A to the two circles meet the circles again at B and C, respectively. Let the point P be located so that AOPQ is a parallelogram. Prove that P is the circumcentre of the triangle ABC.
Q. 16. if two circles intersect in two distinct points A and B and line AB intersects the two common tangents at points P and Q. prove that PA = QB
Q. 17. If PA and PB are two tangents drawn from a point P to a circle with centre O touching it at A and B, prove that OP is the perpendicular bisector of AB.
Q. 18. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC
Q. 19. In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length.
Q. 20. AB is a line segment and M is its mid point .semicircles are drawn with AM, MB and AB as diameters on the same side of the line AB. A circle is drawn to touch all the three semicircles. Prove that its radius r is given by r = 1/6 AB
Q. 21. Two circles touch externally at a point P. from a point T on the tangent at P, tangents TQ and TR drawn to the circles with points of contact Q and R respectively. Prove that TQ = TR.
Q. 22. TA and TB are tangent segments to a circle with centre O, from an external point T. if OT intersects the circle in P, prove that AP bisects 
Q. 23. Two circles intersect at two points A and B and a straight line PAQ intersects the circles at P and Q. if the tangents at P and Q intersect in T, prove that P, B, Q, T are concyclic.
Q. 24. The incircle of ABC touches the sides BC, CA and AB at D, E, and F respectively. Prove that AF + BD + CE = AE + BF + CD = ½ Perimeter of 
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