I proved this theorem in the New Year's night of December 31, 1999. So this is perhaps the first theorem of the year 2000 (published on January 1, 2000).
The text appeared in the renowned Dutch magazine Pythagoras:
https://pyth.eu/uploads/user/ArchiefPDF/Pyth39-5.pdf.

THEOREM.

Let ABC be an equilateral triangle and P an arbitrary point inside this triangle. The points P_1, P₂ and P₃ are the orthogonal projections of P on the sides AB, BC and CA.
The points R, S and T are the centers of the circumscribed circles of the (cyclic) quadrilaterals PP₁AP₃, PP₁BP₂ and PP₂CP₃. Then the area of the triangle RST is one fourth of the area of the given triangle ABC.

Notice that this theorem is also true for an arbitrary triangle ABC and in the case that P lies outside the triangle.
In addition, the theorem is true for an arbitrary quadrilateral, an arbitrary polygon with n sides and in the limit case also for a circle.

Hint for the proof: apply a homothety with center P and ratio ½. You can also use the argument that RS = ½ · AB (midsegment in a triangle).