Question

Find that point equidistant from the four points (a,0,0),(0,b,0),(0,0,c) and (0,0,0).

Answer 1

The point P(a/2, b/2, c/2) is equidistant from O(0,0,0), A(a,0,0), B(0,b,0), and C(0,0,c).

Take triangle OAB. Circumcenter D = midpoint of hypotenuse AB as it's a right angle triangle. D is equidistant from O A and B.

D = (a/2, b/2, 0). Draw a line L perpendicular to xy plane through D. A point P on this line is: P(a/2,b/2,z).


Any point P Equidistant from A B or C is on line L. Now

OP^2= AP^2= BP^2= CP^2
So. a^2 /4 + b^2 /4 + z^2 = a^2/4 + b^2/4 + (z-c)^2

So z = c/2.

Answer (a/2,b/2,c/2).