The orange circle is a locus of points with constant sum of square distances from the vertices of a pentagon ABCDE; I.e., |GA|2 + |GB|2 + …+ |GE|2 is a constant. What is the center of the circle?
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Solution
It’s the centroid of the five vertices.
As vectors, the average of the vectors A, B, C, D, E is G=(A+B+C+D+E)/5. Square distance sum is (G-A).(G-A) + … + (G-E).(G-E), which expands to 5*G2 – 2*(A+B+C+C+E).G + A2+B2+C2+D2+E2. This is constant.
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