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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?

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  • $\begingroup$ If you want to contain all the points, you need to use the circle with the largest radius (rather than smallest radius). $\endgroup$ Mar 23 '19 at 15:42
  • $\begingroup$ @JacobJones that doesn't give the answer. The region formed will be a trapezium, but I don't know how they got that. $\endgroup$
    – Raghav
    Mar 23 '19 at 15:50
  • $\begingroup$ I misread the question. Start by finding the midpoint between $P_0$ and the other points. I believe these midpoints will form the shape containing the points you are interested in. $\endgroup$ Mar 23 '19 at 15:53
  • $\begingroup$ @JacobJones That doesn't do too. $\endgroup$
    – Raghav
    Mar 23 '19 at 16:00
  • $\begingroup$ You're right, I didn't think about that enough. I believe this solves it though. Find the midpoints between $P_0$ and the other points. 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 $P_i$. $\endgroup$ Mar 23 '19 at 16:09
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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). Image showing the resulting shape.

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