In trapezoid ABCD with bases BC and AD, diagonals AC and BD intersect at point O.

In trapezoid ABCD with bases BC and AD, diagonals AC and BD intersect at point O. Prove the equality of the areas of triangles AOB and COD.

Let’s build the height of the CH trapezoid and consider the triangles ABD and ACD, which have a common base AD and the total height of CH, then the areas of triangles ABD and ACD are equal.

Savd = Sasd.

The area of the triangle ABD consists of the sum of the areas of the triangles AOB and AOD.

Saod = Savo + Saod

Similarly Sasd = Ssod + Sаod.

Then Sао + Sаod = Sсod + Sаod, and therefore Sаоv = Sаod, as required.