Given: Points are O(0, 0, 0), A(a, 0, 0), B(0, b, 0) and C(0, 0, c)

To find: the coordinates of point which is equidistant from the points

Let required point P(x, y, z)

According to question:

PA = PB = PC = PO

⇒ PA² = PB² = PC² = PO²

Formula used:

Distance between any two points (a, b, c) and (m, n, o) is given by,

Therefore,

The distance between P(x, y, z) and O(0, 0, 0) is PO,

Distance between P(x, y, z) and A(a, 0, 0) is PA,

Distance between P(x, y, z) and B(0, b, 0) is PB,

Distance between P(x, y, z) and C(0, 0, c) is PC,

As PO² = PA²

x²+ y² + z² = (x – a)² + y² + z²

⇒ x²= x²+ a² – 2ax

⇒ 2ax = a²

As PO² = PB²

x²+ y² + z² = x²+ (y – b)² + z²

⇒ y²= y²+ b² – 2by

⇒ 2by = b²

As PO² = PC²

x²+ y² + z² = x² + y² + (z – c)²

⇒ z²= z²+ c² – 2cz

⇒ 2cz = c²

Hence point is equidistant from given points