Diagonals `A C` and `B D` of a trapezium `A B C D` with `A B D C` intersect each other at the point `O` . Using similarity criterion for two triangles, show that `(O A)/(O C)=(O B)/(O D)` .

Diagonals AC and BD of a trapezium ABCD with AB|DC intersect each other at the point O.Using a similarity criterion for two triangles,show that (OA)/(OC)=(OB)/(OD)

Diagonals A C and B D of a quadrilateral A B C D intersect at O in such a way that a r( A O D)=a r( B O C) . Prove that A B C D is a trapezium.

If the diagonals A C ,B D of a quadrilateral A B C D , intersect at O , and seqarate the quadrilateral into four triangles of equal area, show that quadrilateral A B C D is a parallelogram. GIVEN : A quadrilateral A B C D such that its diagonals A C and B D intersect at O and separate it into four parts such that a r( A O B)=a r( B O C)=a r( C O D)=a r( A O D) TO PROVE : Quadrilateral A B C D is a parallelogram.

Diagonals A C a n d B D of a quadrilateral A B C D intersect at O in such a way that a r ( A O D)=a r ( B O C)dot Prove that A B C D is a trapezium.

In an isosceles triangle ABC, with AB = A C , the bisectors of B and C intersect each other at O. Join A to O. Show that : (i) O B = O C (ii) AO bisects A

The diagonals of quadrilateral A B C D , A C a n d B D intersect in Odot Prove that if B O=O D , the triangles A B C a n d A D C are equal in area.

In Figure, A B C D is a trapezium in which A B||C D . Prove that: a r( A O D)=a r( B O C)dot

In figure ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that (a r(A B C))/(a r(D B C))=(A O)/(D O) .

A point O is taken inside an equilateral four sided figure A B C D such that its distances from the angular points D and B are equal. Show that A O and O C are in one and the same straight line. GIVEN : A point O inside an equilateral quadrilateral four sided figure A B C D such that B O=O Ddot TO PROVE : A O and O C are in one and the same straight line.

In Fig. 9.25, diagonals AC and BD of quadrilateral ABCD intersect at O such that O B" "=" "O D . If A B" "=" "C D , then show that: (i) a r" "(D O C)" "=" "a r" "(A O B) (ii) a r" "(D C B)" "=" "a r" "(A C B) (iii) D A" "||" "C