Two circles `C_1` and `C_2` intersect at two distinct points `P`and`Q` in a line passing through `P` meets circles `C_1`and`C_2` at `A`and`B` , respectively. Let `Y` be the midpoint of `A B` and `Q Y` meets circles `C_1` and`C_2` at `X`a n d`Z` respectively. Then prove that `Y` is the midpoint of `X Z`

Two circles C_1a n dC_2 intersect at two distinct points Pa n dQ in a line passing through P meets circles C_1a n dC_2 at Aa n dB , respectively. Let Y be the midpoint of A B ,a n dQ Y meets circles C_1a n dC_2 at Xa n dZ , respectively. Then prove that Y is the midpoint of X Zdot

Two circles intersects in points P and Q. A secant passing through P intersects the circles in A and B respectively. Tangents to the circles at A and B intersect at T. Prove that the A,Q,B and T lie on a circle.

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig. 10.40). Prove that /_A C P\ =/_Q C D .

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Figure). Prove that /_A C P=/_Q C D .

Two equal circles intersect in P and Q . A straight line through P meets the circles in A and B . Prove that Q A=Q Bdot

Two equal circles intersect in P and Q . A straight line through P meets the circles in A and B . Prove that Q A=Q Bdot

Consider two lines L_1a n dL_2 given by a_1x+b_1y+c_1=0a n da_2x+b_2y+c_2=0 respectively where c1 and c2 !=0, intersecting at point PdotA line L_3 is drawn through the origin meeting the lines L_1a n dL_2 at Aa n dB , respectively, such that PA=P B . Similarly, one more line L_4 is drawn through the origin meeting the lines L_1a n dL_2 at A_1a n dB_2, respectively, such that P A_1=P B_1dot Obtain the combined equation of lines L_3a n dL_4dot

P ,Q and R are, respectively, the mid-points of sides B C ,C A and A B of a triangle A B C , P R and B Q meet at XdotC R and P Q meet at Y . Prove that X Y=1/4B Cdot

The line y=mx+c intersects the circle x^(2)+y^(2)=r^(2) in two distinct points if

A circle having centre at C is made to pass through the point P(1,2), touching the straight lines 7x-y=5 and x+y+13=0 at A and B respectively. Then