At the endpoint and midpoint of a circular are AB, tangent lines are drawn, and the points, A and B are jointed with a chord. Prove that the ratio of the areas of the triangles thus formed tends to 4 as the arc AB decreases infinitely.

Points A, B, C lie on the parabola y^2=4ax The tangents to the parabola at A, B and C, taken in pair, intersect at points P, Q and R. Determine the ratio of the areas of the triangle ABC and triangle PQR

ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively intersecting at P, Q and R. Prove that the perimeter of DeltaPQR is double the perimeter of DeltaABC.

Find the area of a triangle formed by the points A(5, 2), B(4, 7) and C(7, -4).

In a right triangle ABC, a circle with a side. AB as diameter is drawn to intersect the hypotenuse AC in P. Prove that the tangent to the circle at P bisects the side BC.

Two radii OA and OB of a circle with centre O are perpendicular to each other. If two tangents are drawn at the point A and B intersect each other at the point T, prove that AB= OT and they bisect each other at a ridght angle.

(b) (iii) Find the area of the triangle formed by joining the points (4,8),(-4,3) and (12,2) successively.

AB is a diameter of a circle with centre O . A line drawn through the point A intersects the circle at the point C and the tangent through B at the point D . Prove that (a) BD^(2)=AD.DC . (b) The area of the rectangle of formed by AC and AD for any straight line is always equal.

Tangents are drawn to the parabola at three distinct points. Prove that these tangent lines always make a triangle and that the locus of the orthocentre of the triangle is the directrix of the parabola.

The sides of triangle ABC satisfy the relations a + b - c= 2 and 2ab -c^(2) =4 , then the square of the area of triangle is ______