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`

A B C D is a parallelogram and E\ a n d\ F are the centroids of triangles A B D\ a n d\ B C D respectively, then E F= (a) A E (b) B E (c) C E (d) D E

In Figure A B C D\ a n d\ A E F D are two parallelograms. Prove that: P E=F Q

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

A B C D is a parallelogram, E\ a n d\ F are the mid-points of A B\ a n d\ C D respectively. G H is any line intersecting A D ,\ E F\ a n d\ B C at G ,\ P\ a n d\ H respectively. Prove that G P=P H

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)

If E ,F ,G and 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)dot

D , E , F are the foot of the perpendicular from vertices A , B , C to sides B C ,\ C A ,\ A B respectively, and H is the orthocentre of acute angled triangle A B C ; where a , b , c are the sides of triangle A B C , then H is the incentre of triangle D E F A , B ,\ C are excentres of triangle D E F Perimeter of D E F is a cos A+b cos B+c\ cos C Circumradius of triangle D E F is R/2, where R is circumradius of A B C

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))

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

A B C D is a parallelogram, G is the point on A B such that A G=2\ G B ,\ E\ is a point of D C such that C E=2D E\ a n d\ F is the point of B C such that B F=2F C . Prove that: a r\ (triangle \ E F C)=1/2\ a r\ (triangle \ E B F)