Calculus: Find equation that represents the set of all points that are equidistant from given three points (0,0,0) (2,4,3)(10,8,9)

Question

So it is essential circumcenter problem in 3D that involves multivariable calculus. If you could at least help with the ideas or steps of tackling this problem that would be great

PS: I made up the points, if somehow they do not work feel free to use other three points in the space.

Thanks!

Answer 1

You need to solve $$ x^2+y^2+z^2=(x-2)^2+(y-4)^2+(z-3)^2=(x-10)^2+(y-8)^2+(z-9)^2, $$ each equation is the distance (squared) from $(x,y,z)$ to your three points.

Answer 2

Procedure. Given points $P,Q,R$, take two of the points, say $P,Q$, and construct (describe, find an equation) of the plane that passes through their midpoint, and is perpendicular to the line segment connecting them. Take another pair of points, say $P,R$ and repeat the procedure. Intersect the two planes obtained in the previous steps. Obtain a line, which is the set of all points equidistant from $P,Q,R$.