Consider $6$ points located at $P_0=(0,0), P_1=(0,4), P_2=(4,0), P_3=(-2,-2), P_4=(3,3), P_5=(5,5)$. Let $R$ be the region consisting of all points in the plane whose distance from $P_0$ is smaller than that from any other $P_i$, $i=1,2,3,4,5$. Find the perimeter of the region $R$.
I thought of calculating the the circumference of circle with lowest radius from the given points, but apparently it doesn't gives the answer. Where am I going wrong?
Find the midpoints between $P_0$ and the other points (dark blue in the image). At that midpoint, draw a line perpendicular to the line from $P_0$ to the corresponding point. The shape traced by these perpendicular lines contains all points closer to $P_0$ than to another $P_i$ (the light green shaded region).