Some years ago, I asked some famous people in Advanced Plane Geometry forum that:

> *Let $ABC$ be arbitrary triangle, how can construct point $P$ in the plane such that $P$ is the circumcenter of the cevian triangle of $P$ respect to $ABC$*

The answer is negative, even can not calculation by a Computer. 

> **My conjecture:** *Let $ABC$ be arbitrary triangle, then always exist point $P$ such that $P$ is the circumcenter of the cevian triangle of $P$ respect to $ABC$*

I proposed the conjecture to You.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/a4W0L.png