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head_hunter

Circumcenter calculation

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Hi everyone. I am trying to calculate the circumcenter of a triangle in 3D space. The way I see it, the geometric properties of the circumcenter can be used to compute its coordinates. The property I have in mind is that the circumcenter of a triangle is the point of intersection of all the three lines that: a) are perpendicular to the triangle's sides and b) intersect the triangle's sides at their midpoint This means that if A, B and C are the triangle's vertices, O the circumcenter and: 1) D the midpoint of A, B 2) F the midpoint of B, C 3) G the midpoint of C, A then if must hold that: (O - D) dot (A - B) = 0 (O - F) dot (B - C) = 0 (O - G) dot (C - A) = 0 The three equations above have 3 unknowns each (the x, y and z coordinates of the circumcenter O) and, therefore, present a solvable system of linear equations. The problem is that the coefficient matrix of the above system has determinant equal to 0 and cannot be inverted. Can someone please explain why this is so and perhaps present an algorithm for the calculation of a 3D triangle's cicrumcenter? Thanks.

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