Prove that any three tangents to a parabola whose slopes are in harmonic progression enclose a triangle of constant area.

Prove that the normal chord to a parabola at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.

Prove that the equation of the parabola whose vertex and focus are on the X-axis at a distance a and a'from the origin respectively is y^2=4(a'-a)(x-a)

Prove that the semi-latus rectum of the parabola 'y^2 = 4ax' is the harmonic mean between the segments of any focal chord of the parabola.

TP and TQ are any two tangents to a parabola and the tangent at a third point R cuts them in P' and Q' . Prove that (TP')/(TP)+(TQ')/(TQ)=1

Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.

Prove that an equilateral triangle can be constructed on any given line segment.

Prove that two triangles on the same base and between the same parallels are equal in area.

The normals at three points P,Q,R of the parabola y^2=4ax meet in (h,k) The centroid of triangle PQR lies on

Prove that the equation y^2+2ax+2by+c=0 represent a parabola whose axis is parallel to the axis of x. Find its vertex.

Prove that area at triangle will remain same by shifting origin at any point.