Question

ABCD is a quadrilateral in which diagonals AC and BD intersect at a point O. Prove that: areaΔAOD + areaΔBOC + areaΔABO + areaΔCDO.

Answer 1

Since the diagonals of a parallelogram bisect each other at the point of intersection.
Therefore, OB = OD and OA = OC
In ΔABC, OB is the median and median divides triangle into two triangles of equal areas
Therefore, area(ΔBOC) = area(ΔABO)   ..........(i)
In ΔADC, OD is the median and median divides triangle into the triangles of equal areas
Therefore, area(ΔAOD) = area(ΔCDO)  ..........(ii)
Adding (i) and (ii)
area(ΔAOD) + area(ΔBOC) = area(ΔABO) + area(ΔCDO).