You could begin by saying that the areas of triangles OSR and SOE are equal

as they have the same base and perpendicular heights.

This means that triangles OIR and SIE also have equal areas as they differ from the above-mentioned triangles

by the area of triangle OIS.

Now write the area of the triangle OIR using the angle at I

Do the same for triangle SIE

However, those angles are identical.

Hence the result follows