The L point of a four-sided polygon.

Let ABC be a triangle and K the circumcircle of it. Take any point P of K. It is known that the projections P_AB, P_BC and P_CA of this point on the three edges of the triangle are exactly on a line G.

I extended this to quadrilaterals inscribed in a circle:

Take a quadiralteral ABCD with the four vertices on a circle. We have the four triangles ABC, ACD, ABD and BCD.

Take for the triangle ABC the point D. From the previous geometrical theorem we know that the projections of D on the triangle edges are on a line G_D.

Take for the triangle ACD and the point B we obtain similar the line G_B. In the same way we obtain the lines G_A and G_C

Now my theorem states that the four lines G_A, G_B, G_C, and G_D have a common intersection point L.

Prove that this can not be extended to polygons with more than 4 sides!

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