In ` A B C ,/_B=2/_C` and the bisector of `/_B` intersects `A C` at `Ddot` Prove that `(B D)/(D A)=(B C)/(B A)`

In a parallelogram A B C D , the bisector of /_A also bisects B C at Xdot Prove that A D=2A B

A B C D is a trapezium with A B D Cdot A line parallel to A C intersects A B at X and B C at Ydot Prove that a r( A D X)=a r( A C Y)dot

In a parallelogram A B C D ,\ the bisector of /_A also bisects B C at Xdot Find A B : A D

In Figure, A B C is an isosceles triangle in which A B=A CdotC P A B and A P is the bisector of exterior /_C A D of A B C . Prove that /_P A C=/_B C A and (ii) A B C P is a parallelogram.

In Fig. 4.179, A B C and D B C are on the same base B C . If A D and B C intersect at O , prove that (A r e a\ ( A B C))/(A r e a\ ( D B C))=(A O)/(D O) (FIGURE)

In figure A B=A C\ a n d\ C P||B A and A P is the bisector of exterior /_C A D of A B C . Prove that /_P A C=/_B C A (ii) A B C P is a parallelogram

A B C D is a trapezium with A B||D C . A line parallel to A C intersects A B at X a n d B C at Ydot Prove that a r( A D X)=a r( A C Y)dot

O is any point on the diagonal B D of the parallelogram A B C Ddot Prove that a r( O A B)=a r( O B C)

D is the mid-point of side B C of A B C and E is the mid-point of B Ddot If O is the mid-point of A E , prove that a r( B O E)=1/8a r( A B C)

The diagonals of quadrilateral A B C D ,A C and B D intersect in Odot Prove that if B O=O D , the triangles A B C and A D C are equal in area. GIVEN : A quadrilateral A B C E in which its diagonals A C and B D intersect at O such that B O = O Ddot TO PROVE : a r( A B C)=a r( A D C)