This is a problem of solving equations. Formulate a system of equations based on known conditions. Express x and y using known conditions. What you get is the formula you want

Lemma

Suppose $A$, $B$ are two points on the plane, $lambda > 0, lambda neq 1$, the trajectory of $P$ point satisfying $frac{|AP|}{|BP|}=lambda$ is A circle (Apollonian circle); when $lambda = 1$, the trajectory degenerates into a straight line, the perpendicular to $AB$. When not degenerate, the circle has $CD$ as its diameter, where $C, D$ are located on the straight line $AB$, and satisfy $C$ on the line segment $AB$, $D$ outside the line segment $AB$, $frac {|AC|}{|BC|}=frac{|AD|}{|BD|}=lambda$.

According to this lemma, you first select $P_1$, $P_2$ and get a trajectory $Gamma_1$; then select $P_2$, $P_3$ and get the second trajectory $Gamma_2$. If $Gamma_1,Gamma_2$ has an intersection point, the intersection point is the desired one.

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Determine whether a point is within a triangle on a two-dimensional plane

As for finding the ratio from M point to each point, just find the distance to each point.

Solution of quadratic equation in two variables