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(1, -1), B(4, - 2) and C(9,3) be the vertices of the triangle ABC . A parallelogram AFDE is drawn with vertices D, E and F on the line segments BC, CA and AB respectively Find the maximum area of parallelogram AFDE .

If P(1,2), Q(4,6), R(5,7) and S(a, b) are the vertices of a parallelogram P Q R S , then

p A N D q are any two points lying on the sides D C a n d A D respectively of a parallelogram A B C Ddot Show that a r( A P B)=a r ( B Q C)dot

P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD . Show that a r\ (A P B)\ =\ a r\ (B Q C) .

Let P be an interior point of a triangle ABC , Let Q and R be the reflections of P in AB and AC , respectively if Q .A , R are collinear then angle A equals -

Let P be an interior point of a triangle ABC , Let Q and R be the reflections of P in AB and AC , respectively if Q .A , R are collinear then angle A equals -

Let P be an interior point of a triangle ABC , Let Q and R be the reflections of P in AB and AC , respectively if Q .A , R are collinear then angle A equals -

Prove that the figure formed by joining the mid-points of the pair of consecutive sides of a quadrilateral is a parallelogram. OR ABC D is a parallelogram in which P ,Q ,R and S are mid-points of the sides A B ,B C ,C D and D A respectively. A C is a diagonal. Show that : P Q|| A C and P Q=1/2A C S R || A C and S R=1/2A C P Q=S R (iv) P Q R S is a parallelogram.

Prove that the figure formed by joining the mid-points of the pair of consecutive sides of a quadrilateral is a parallelogram. OR B A C D is a parallelogram in which P ,Q ,R and S are mid-points of the sides A B ,B C ,C D and D A respectively. A C is a diagonal. Show that : P Q|| A C and P Q=1/2A C S R || A C and S R=1/2A C P Q=S R (iv) P Q R S is a parallelogram.

p\ A N D\ q are any two points lying on the sides D C\ a n d\ A D respectively of a parallelogram A B C Ddot\ Show that a r( A P B)=a r\ ( B Q C)dot