If A,B,C are three points on a line and B lies between A and C, then prove that AC-AB = BC

PQ is a diameter of a circle and AB is such a chord of it that it is perpendicular to PQ . If C be the point of intersection of PQ and AB , then prove that PC.QC=AC.BC .

If a , b ,c are three distinct positive real numbers in G.P., then prove that c^2+2a b >3ac

Let A,B,C be three unit vectors and A.B= A.C=0. If the angle between B and C is pi/6 , then A is equals to

A straight line intersects one of the two concentric circles at the points A and B and other at the points C and D. Prove that AC = BD.

In the adjacent figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC||PR. Show that BC || QR.

If the polynomial (a^(2)+b^(2)+c^(2)-ab-bc-ca)x^(2)+2abcx+(a^(2)+b^(2)+c^(2)) , where a, b, c are constants, be a linear polynomial, then prove that a = b = c.

Two tangents AB and AC drawn from an external point A of a circle touch the circle at the point B and C .A tangent drawn a point X lies on minor are BC intersects AB and AC at the points D and E respectively. Prove that perimeter of DeltaADE=2AB.

If C is a point on the line segment joining A (-3, 4) and B (2, 1) such that AC=2BC, then the coordinate of C is

Two parallel straight lines intersect three concurrent straight lines at the points A , B , C and X,Y,Z respectively. Prove that AB : BC=XY : YZ .

Two tangents AB and AC drawn from an external point A of a circle touch the circle at the point B and C. A tangent drawn to a point X lies on minor arc BC intersects AB and AC at the points D and E respectively. Prove that peremeter of triangleADE = 2AB .