Write the equation $f(x,y)=0$ that describes the set $S$ Let $A=(1,1)$, $B=(1, -2)$ and $C=(5,-5)$.
Write the equation $f(x,y)=0$ with $f\in \mathbb{R}[x,y]$ that describes the set $S$ of points that have same distance of $A$ as to $BC$.
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I have done the following:
Let $P=(x,y)$ be a point of $S$.
The it must be $d(P,A)=d(P,BC)$.
The line through $B$ and $C$ is $$y+2=\frac{-2+5}{1-5}(x-1) \Rightarrow y+2=-\frac{3}{4}(x-1) \Rightarrow 4y+8=-3x+3 \Rightarrow 3x+4y+5=0$$
From $d(P,A)=d(P,BC)$ we get $$\sqrt{(x-1)^2+(y-1)^2}=\frac{|3x+4y+5|}{\sqrt{3^2+4^2}}$$ right?
Do we get from that the desired equation?

We have
$$\sqrt{(x-1)^2+(y-1)^2}=\frac{|3x+4y+5|}{\sqrt{3^2+4^2}} \iff 25(x-1)^2+25(y-1)^2-(3x+4y+5)^2=0.$$
