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**Richard** Given Euclidean geometry, how to generate for any pair of distinct points $\displaystyle x$ and $\displaystyle y$, for any factor $\displaystyle 0 \leq \lambda \leq 1$, two halfplanes $\displaystyle X$ and $\displaystyle Y$, where $\displaystyle X$ consists of all points $\displaystyle v$ with $\displaystyle \lambda d(v,x) < (1 - \lambda) d(v,y)$? In the simple case where $\displaystyle \lambda = 1 - \lambda = .5 $, it's easy. You just draw a line through $\displaystyle x$ and $\displaystyle y$, half it and draw the perpendicular line through the center of the line. But how do you do it otherwise?