$M$ is the intersection of 3 cevians in the triangle $ABC$.

$$AB_1 = x,\quad CA_1 = y,\quad BC_1= z.$$

It can be easily proven that for both *Nagel* and *Gergonne* points the following equation is true:
$$S = xyz / r,$$
where $S$ is the area of the triangle $ABC$ and $r$ is the radius of the inscribed circle.

I wonder what other triangle centers might possibly have the same property and what is the geometric place for them?

Also, please note that for the case where point $M$ is the centroid the formula looks as follows: $S = 2xyz/R$, where $R$ is the radius of the circumcircle. Substitution $x = b/2$, $y = a/2$, $z = c/2$ brings it back to the classic $S = abc/4R$. Perhaps, some other triangle centers might exist, so that this equation $S = 2xyz/R$ holds true for them as well. I wonder in what particular relation these hypothetical points might be to the centroid of $ABC$?

S = xyz/r, which reads badly (e.g., compare $a$`$a$`

vs.a`*a*`

). I have edited accordingly. $\endgroup$ – LSpice Aug 22 '20 at 13:07