I've been working on a geometry/trig problem in my spare time, which is concisely displayed here: https://www.desmos.com/calculator/tt5sh99igg. If you don't want to look at the desmos link (which you don't have to), I basically have ∆ABC and random Point P (all of which are on a normal cartesian coordinate plane). I've thought of a few things that I'd like to derive from these givens, but the one I'm working on right now is this: Express all Points $P$ for any $\Delta ABC$ where $PA+PB+PC$ equals the perimeter of $\Delta ABC$. I had been trying to look for an alternative other than painful painful algebra, but unfortunately I haven't found one yet, so I decided to brute force the algebra. Here's the beast:
$$\sqrt{\left ( a_x-b_x \right )^2 + (a_y-b_y)^2}+\sqrt{\left ( b_x-c_x \right )^2 + (b_y-c_y)^2}+\sqrt{\left ( c_x-a_x \right )^2 + (c_y-a_y)^2}=\sqrt{\left ( a_x-x \right )^2 + (a_y-y)^2}+\sqrt{\left ( b_x-x \right )^2 + (b_y-y)^2}+\sqrt{\left ( c_x-x \right )^2 + (c_y-y)^2}$$
So, I have a few questions. Is it possible to solve this (in terms of $y$)? Are there other non-brute force methods to either solve this equation or derive the solution I'm looking for? Is there a way to simplify this (in addition to having point (ax, ay) = (0, 0))? And lastly, if you happen to think of anything else that would be fun or interesting to derive with these givens, please suggest below!
EDIT (for clarification): $x$ and $y$ are unknown variables. The variables ax, ay, bx, by, cx, and cy are all known constants.
EDIT 2: A user commented that a combination of three equations for part of an ellipse would work. Or in other words, three functions of an ellipse, one with foci at points $A$ and $B$, another with foci at $B$ and $C$, and the last with foci at points $C$ and $A$. After testing this myself, I've concluded this isn't correct. In addition to using a paper which @Kajelad linked in the comments section, might there be any other methods of finding a solution?