Let n>2 be a natural number and a=cos(2pi/n) + isin(2pi/n). Let i,j,k be three distinct elements of set {0,1,2,...,n-1}. Find all complex numbers z satisfying /z-a^i/=/z-a^j/=/z-a^k/, where /u/ signifies the modulus of complex number u.
Since a=cos(2pi/n) + isin(2pi/n) and i,j,k are in {0,1,...,n-1} and distinct, ai, aj, and ak are three distinct nth roots of unity. Geometrically, ai, aj, and ak are three distinct points on the unit circle defining a triangle whose circumcenter is the only point that is equidistant from the three vertices of the triangle. Hence, z is the circumcenter of the triangle whose vertices are ai, aj, and ak. To find the coordinates of z, we may use the formula given here https://www.geeksforgeeks.org/circumcenter-of-triangle/.
Spiros Konstantogiannis Very good answer, correct solution and......z=o, because the three points are on the unit circle having the centre in the origin!
Dear Dinu, you are right. Actually, I had forgotten that the circumcenter of a triangle may be outside of it, or on one of its sides, and for this reason, I had not realized the obvious fact you mentioned.
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