Center of circle passing through any 3 points

In 2023 alevel P2, one of the challenging question is to find the center of circle passing through 3 points. The 3 points form an isosceles triangle. Hence the center of circle will lie on a line passing through one of the vertices.

Alevel has the tendency of making the question harder in subsequent years, just like 2018 probability bug question becomes 2022 probability counter question. So students must be prepared to find the center of circle passing through any 3 points.

Practice question: Find the center of circle passing through (1,2,3), (2,1,4) and (3,4,5)

Answer: (17/8,11/4,33/8)

Hints

Let the 3 points be A, B, C and the center of circle be P (x,y,z)

Find the cartesian equation of plane ABC: -x+z=2

Since P lies on plane ABC, P fulfill equation of plane ABC, -x+z= 2 (Equation 1)

AP=BP (radius of circle)

(Steps are left out for students to try)

This reduces to 2x-2y+2z=7 (Equation 2)

AP=CP (radius of circle)

(Steps are left out for students to try)

This reduces to x+y+z=9 (Equation 3)

Solving the 3 equations, this gives x= 17/8 , y=11/4, z=33/8

Hence coordinates of P = (17/8,11/4,33/8)