ABCD is a trapezium in which AB || CD and its diagonals intersect each other at the point O.

show that; \frac{AO}{BO}=\frac{CO}{DO}
show that; \frac{AO}{BO}=\frac{CO}{DO}
Step 1: Construct EF parallel to AB and CD
NOTE: EF || AB || CD
Step 2: Apply the basic proportional theorem to triangle ACD and ACB
THEOREM: If a line is drawn parallel to one side of a triangle to intersect
the other two sides in distinct points, then the other two sides are divided
in the same ratio.
EXAMPLE: In ∆ACD, EO || CD
\frac{AO}{OC}=\frac{EA}{ED}...................(1)
In ∆ACB, EO || AB
\frac{DO}{OB}=\frac{ED}{EA}
Take the reciprocal on both sides
\frac{BO}{DO}=\frac{EA}{ED} ........................(2)
Step 3: Prove the required ratio by using equation (1) and (2)
From equation (1) and (2) we can write
\frac{AO}{OC}=\frac{BO}{DO}
\frac{AO}{BO}=\frac{OC}{OD}
Hence proved