Locus of points with distances from lines

Question

Given two lines, find the locus of points whose distance from the first line is two times the distance from the second line. I prefer a solution with Euclidean geometry. With analytic it's quite easy

Answer 1

If your lines $r,s$ form an angle $\alpha<\pi$, consider a triangle $RPS$ in which $R\in r,\widehat{RPS}=\pi-\alpha$, $PR=2,PS=1$ and $PR\perp r$. By "sliding" $R$ over $r$ you can assume that $S\in s$ and hence $PS\perp s$. Now, by similarity, any point $Q$ on the $OP$ line, where $O=r\cap s$, has the property that $d(Q,r)=2\cdot d(Q,s)$. You only have to prove that no other points have this property, that is quite easy.