Prove that if you change the quadrilateral, keeping the same lengths of its diagonals and the angle between

Prove that if you change the quadrilateral, keeping the same lengths of its diagonals and the angle between them, then the area of the quadrilateral will not change.

The intersection point of the diagonals O of the quadrilateral ABCD is the vertex of the angles α and β of triangles with areas:

½ * AO * BO * sinα for ∆ AOB;

½ * BO * CO * sinβ for ∆ BOC;

½ * CO * DO * sinα for ∆ COD;

½ * AO * DO * sinβ for ∆ AOD;

Where:

sinβ = sin (180 ° – α) = sinα;

Adding these areas together, we find the area ABCD:

S = ½ * (AO * BO + BO * CO + CO * DO + AO * DO) * sinα = ½ * AC * BD * sinα;

Quadrangles with the same diagonals and the angle α between them have the same area, which is what was required to prove.