The geometric place (locus) of the point $P$ is the incircle of $\triangle ABC$. Found by applying analytical geometry.

As above, we claim the locus is the incircle. Let $D,E$ be its contact points with $CA, AB$ respectively, i.e. $DE$ is midline of the triangle, and $P$ be a point of the smaller arc $DE$ of the incircle. We note $K,L,M,N$ the projections of $P$ onto $BC, CA, AB, DE$ respectively. For convenience $KP=x, PL=y, MP=z, PN=m, KN=n$. 1) Well known property: the square of the distance of a point belonging to a circle to any of its chords equals the product of distances of the same point to the tangents at the endpoints of the chord to the circle, hence $m^2=yz\ (\ 1\ )$. This is easy to be proven, seeing that $\triangle MNP\sim\triangle NLP$. 2) Other well known property: the sum of the distances of any inner point of an equilateral triangle to its sides equals the altitude of the triangle; since $DE$ is a midline, we may say $m+x+y=n\ (\ 2\ )\implies x=2m+y+z\ (\ 3\ )$. From the relations $(3)$ and $(1)$ we get the required $\sqrt x=\sqrt y+\sqrt z$. Best regards, sunken rock

sunken rock wrote: As above, we claim the locus is the incircle. Let $D,E$ be its contact points with $CA, AB$ respectively, i.e. $DE$ is midline of the triangle, and $P$ be a point of the smaller arc $DE$ of the incircle. We note $K,L,M,N$ the projections of $P$ onto $BC, CA, AB, DE$ respectively. For convenience $KP=x, PL=y, MP=z, PN=m, KN=n$. 1) Well known property: the square of the distance of a point belonging to a circle to any of its chords equals the product of distances of the same point to the tangents at the endpoints of the chord to the circle, hence $m^2=yz\ (\ 1\ )$. This is easy to be proven, seeing that $\triangle MNP\sim\triangle NLP$. 2) Other well known property: the sum of the distances of any inner point of an equilateral triangle to its sides equals the altitude of the triangle; since $DE$ is a midline, we may say $m+x+y=n\ (\ 2\ )\implies x=2m+y+z\ (\ 3\ )$. From the relations $(3)$ and $(1)$ we get the required $\sqrt x=\sqrt y+\sqrt z$. Best regards, sunken rock Actually, the equation $(2)$ is $m+y+z=n$.

What about the reciprocal affirmation: "If $\sqrt x=\sqrt y+\sqrt z$, then $P$ belongs to the incircle"?