Two triangle ABC and `D B C` lie on the same side of the base `B C` . From a point `P` on BC, PQ||AB and PR||BD are drawn. They meet `A C` in `Q` and `D C` in `R` respectively. Prove that QR||AD`.

A D is a median of Delta A B C . The bisector of /_A D B and /_A D C meet AB and AC in E and F respectively. Prove that EF||BC

Two towns A and B are situated on the same side of a straight road at distances a and brespectively perpendiculars drawn from A and B meet the road at point C and D respectively.The distance between C and D is c. A hospital is to be built at a point P on the road such thatthe distance APB is minimum. Find position of P.

Two triangles B A Ca n dB D C , right angled at Aa n dD respectively, are drawn on the same base B C and on the same side of B C . If A C and D B intersect at P , prove that A P*P C=D P*P Bdot

Triangles A B C and DBC are on the same base B C with A, D on opposite side of line B C , such that a r(triangle A B C)=a r( triangle D B C) . Show that B C bisects A D .

Triangles A B C and DBC are on the same base B C with A, D on opposite side of line B C , such that a r(_|_ A B C)=a r( D B C)dot Show that B C bisects A Ddot

The bisectors of the angles B and C of a triangle A B C , meet the opposite sides in D and E respectively. If DE||BC , prove that the triangle is isosceles.

A B C D is a rhombus and P ,\ Q ,\ R ,\ S are the mid-points of A B ,\ B C ,\ C D ,\ D A respectively. Prove that P Q R S is a rectangle.

The side B C of a triangle A B C is bisected at D ; O is any point in A D. BO and CO produced meet AC and AB in E and F respectively and A D is produced to X so that D is the mid-point of O X . Prove that A O : A X=A F : A B and show that FE||BC .

If BM and CN are the perpendiculars drawn on the sides AC and BC of the Delta ABC , prove that the points B, C, M and N are concyclic.

In Fig. 10.20, the sides A B ,\ B C and C A of triangle A B C touch a circle with centre O and radius r at P ,\ Q and R respectively. (FIGURE) Prove that: 1) AB + CQ = AC + BQ