The **first** and **second** Lemoine circles are well-known to geometers. According to this article the third Lemoine circle has been first discovered by Jean-Pierre Ehrmann in 2002. It is worth noting that the centres of these circles belong to a line that is going through the Lemoine point and the circumcenter of the triangle.

I would like to present the construction of another circle, which I believe might be called **the 4th Lemoine circle**. The idea behind its construction is slightly similar to that proposed by Ehrmann:

*Symmedians AA', BB', CC' intersect each other at the point M (the Lemoine point of the triangle ABC). The circumcircle of the triangle A'B'C' was drawn (also known as the 'symmedial circle'). A'', B'', C'' are the first intersection points of the symmedians with the symmedial circle. Finally, the circumcircles of the triangles A''B''M, B''C''M, A''C''M always intersect the sides of the original triangle ABC at six points, that are conclycic.*

As far as I can tell the point **O** (the center of our six point circle) is not included into the *C.Kimberling's Encyclopedia*, so it must be unknown or uncatalogued. **O** belongs to the line that contains the centres of the first three Lemoine circles, so by my reckoning, the dotted red circle that is shown in the picture above perfectly qualifies for being called **the 4th Lemoine circle.**

Could you please give a synthetic proof of this theorem ?