`A D ,\ B E\ A N D\ C F` are the perpendiculars from the angular points of a ` A B C` upon the opposite sides. The perimeters of the ` D E F\ a n d\ A B C` are in the ratio. `(2r)/R` (b) `r/(2R)` (c) `r/R` (d) `r/(3R)` Where `r` is the inradius and `R` is circum-radius of the ` A B C`

Consider a D E F , the pedal triangle of the A B C such that A-F-B\ a n d\ \ B-D-C are collinear. If H is the incentre of D E F and R_1,\ R_2,\ R_3 are the circumradii of the quadarilaterals A F H E ; B D H F\ a n d\ C E H D respectively, then prove that sumR_1=R+r where R is the circumradius and r is the inradius of A B Cdot

If D ,E ,F are the mid-points of the sides B C ,C Aa n dA B respectively of a triangle A B C , prove by vector method that A r e aof D E F=1/4(a r e aof A B C)dot

In a A B C ,\ E is the mid-point of median A Ddot Show that a r\ ( B E D)=1/4a r\ ( A B C)

If E ,\ F ,\ G\ a n d\ H are respectively the mid-points of the sides of a parallelogram A B C D ,\ Show that a r(E F G H)=1/2a r\ (A B C D)

In triangle A B C , let /_c=pi/2dot If r is the inradius and R is circumradius of the triangle, then 2(r+R) is equal to a+b (b) b+c c+a (d) a+b+c

A point D is taken on the side B C of a \ A B C such that B D=2D Cdot Prove that a r\ (\ A B D)=\ 2\ a r\ (\ A D C)

In triangle A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A P . Prove that: a r\ ( triangle R Q C)=3/8\ a r\ (triangle \ A B C) .

D E F is the triangle formed by joining the points of contact of the incircle with the sides of the triangle A B C ,\ prove that its sides are 2rcosA/2,\ 2rcosB/2\ a n d\ 2rcosC/2 where r is the radius of incircle of A B Cdot its angles are pi/2-A/2,pi/2-B/2a n dpi/2-C/2 its area is (r^2s)/(2R) where ' s ' is the semiperimter and R is the circumradius of the A B C .

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

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 .