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`

In figure, two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R,S and P,Q respectively. Prove that : seg SQ || seg RP.

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

In the figure, two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R,S and P,Q respectively. Prove that : seg SQ || seg RP.

As shown in the adjoining figure , two circles centred at A and B are touching at C. Line passing through C intersects the two circles at M and N respectively. Show that seg AM || seg BN.

The tangent at any point P on the circle x^2+y^2=4 meets the coordinate axes at Aa n dB . Then find the locus of the midpoint of A Bdot

In figure,ABE is a in which AE=BE .Circle passing through A and B intersects AE and BE at D and C respectively.Prove that DC| AB.

Two circles with centres Aa n dB of radii 3c ma n d4c m respectively intersect at two points Ca n dD such that A Ca n dB C are tangents to the two circles. Find the length of the common chord C Ddot

If a, b, c are in H.P. and they are distinct and positive then prove that a^n+c^n gt 2b^n

In Figure, X\ a n d\ Y are the mid-points of A C\ a n d\ A B respectively, Q P\ B C\ a n d\ C Y Q\ a n d\ B X P are straight lines. Prove that a r\ (\ A B P)=\ a r\ (\ A C Q)

Let the base A B of a triangle A B C be fixed and the vertex C lies on a fixed circle of radius rdot Lines through Aa n dB are drawn to intersect C Ba n dC A , respectively, at Ea n dF such that C E: E B=1:2a n dC F : F A=1:2 . If the point of intersection P of these lines lies on the median through A B for all positions of A B , then the locus of P is a circle of radius r/2 a circle of radius 2r a parabola of latus rectum 4r a rectangular hyperbola