Answer:
Let us take,
D(x,y,z) point equidistant from points A(a, 0, 0), B(0, b, 0), C(0, 0, c) and O(0, 0, 0).
∴ AD = OD
Squaring on both sides,
(x – a)2+ (y – 0)2 + (z – 0)2 = (x – 0)2 + (y – 0)2 + (z – 0)2
x2 +2ax + a2 + y2 + z2 = x2 + y2 + z2
a(2x-a) = 0 ;a ≠ 0
x = a/2
∴ BD = OD
Squaring on both sides,
(x – a)2+ (y – 0)2 + (z – 0)2 = (x – 0)2 + (y – 0)2 + (z – 0)2
b(2y-b) = 0 ;b ≠ 0
y= b/2
∴ CD = OD
Squaring on both sides,
(x – a)2+ (y – 0)2 + (z – 0)2 = (x – 0)2 + (y – 0)2 + (z – 0)2
x2 + y2 + z2 + 2cz + c2 = x2 + y2 + z2
c(2z-c) = 0 ;c ≠ 0
z= c/2
Therefore, the point D(a/2,b/2,c/2) is equidistant to points A(a, 0, 0), B(0, b, 0), C(0, 0, c) and O(0, 0, 0).