In trapezoid ABCD with bases BC and AD, the diagonals AC and BD are drawn

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

Consider triangles ABD and ACD formed by the diagonals of the trapezoid.

In triangles ABD and ACD there is a common base AD.

Let’s build the height of the CH trapezoid, which is also the height of the triangles ABD and ACD, then:

Savd = AD * CH / 2.

Sasd = AD * CH / 2.

Savd = Sasd.

Triangle ABD consists of two triangles, AOB and AOD, triangle ACD consists of triangles COD and AOD.

Then Saod = Saov + Saod.

Sсд = Sсod + Sаod.

Saov + Saod = Scod + Saod.

Saov = Ssod.

Triangles AOB and COD are of equal size, which is what was required to be proved.