Find the locus of points $P$ such that $\Delta A_{1}B_{1}C_{1}\sim\Delta A_{2}B_{2}C_{2}$

Question

Let $ABC$ be an triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. and $AP\cap BC=A_{2},BP\cap AC=B_{2},CP\cap AB=C_{2}$,Find the locus of points $P$ such that $\Delta A_{1}B_{1}C_{1}\sim\Delta A_{2}B_{2}C_{2}$

It is clear $P$ is orthocenter is hold,have other point? the simaler problem it help to solev Geomtry

Answer 1

In other terms, you are looking for the locus of points such that the pedal triangle is similar to the cevian triangle. It has been proved by Ehrmann that the orthocenter is the only point with such a property.