In the figure, XY is a line parallel to BC is drawn through A. If BE || CA and CF || BA are drawn to meet XY at E and F respectively. Show that `ar(DeltaABE) = ar (DeltaACF).`

In the figure, ABCD is a quadrilateral. AC is the diagonal and DE || AC and also DE meets BC produced at E. Show that ar(ABCD) = ar (DeltaABE).

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

The lengths of the tangents drawn from the vertices A, B and C to the incicle of DeltaABC are 5, 3 and 2, respectively. If the lengths of the parts of tangents within the triangle which are drawn parallel to the sides BC, CA and AB of the triangle to the incircle are alpha, beta and gamm , respectively, then the value of [alpha + beta + gamma] (where [.] respresents the greatest integer functin) is _____

Let A(p^2,-p),B(q^2, q),C(r^2,-r) be the vertices of triangle ABC. A parallelogram AFDE is drawn with D,E, and F on the line segments BC, CA and AB, respectively. Using calculus, show that the maximum area of such a parallelogram is 1/2(p+q)(q+r)(p-r)dot

Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that vec(AD)+vec(BE)+vec(CF)=vec(0) .

O is any point inside a triangle ABC. The bisector of angle AOB, angle BOC and angle COA meet the sides AB, BC and CA in point D, E and F respectively. Show that AD xx BE xx CF= DB xx EC xx FA

ABCD is quadrilateral with AB parallel to DC. A line drawn parallel to AB meets AD at P and BC at Q. prove that (AP)/(PD) = (BQ)/(QC)

In trapezlum ABCD, AB || DC, E and F are points on non-parallel sides AD and BC respectively, such that EF || AB. Show that (AE)/(ED)=(BF)/(FC) .

In trapezium ABCD, AB || DC. E and F are points on non-parallel sides AD and BC respectively such that EF || AB. Show that (AE)/(ED) = (BF)/(FC)

Let P be an interior point of a triangle ABC and AP, BP, CP meet the sides BC, CA, AB in D, E, F, respectively. Show that (AP)/(PD)=(AF)/(FB)+(AE)/(EC) .