Collection of Maths Problems

Circle interpolation

Task number: 2355

• Hint

  Use coordinates of points $A$, $B$ and $C$ in the circle equation $(x-m)^2+(y-n)^2=r^2$. Subtract the quadratic terms to get the coordinates of the centre $S=(m,n)$.

• Resolution

  We get equations

  $$\begin{eqnarray*} (2-m)^2 + (1-n)^2 = r^2 \\ (4-m)^2 + (3-n)^2 = r^2 \\ (0-m)^2 + (7-n)^2 = r^2 \end{eqnarray*}$$

  After the expansion

  $$\begin{eqnarray*} -4m + m^2 -2n + n^2 + 5 = r^2 \\ -8m+ m^2 -6n + n^2 + 25 = r^2 \\ m^2 -14n+ n^2 + 49 = r^2 \end{eqnarray*}$$

  When we subtract the third from the previous, we get a system of two linear equation with two unknowns:

  $$\begin{eqnarray*} -4m +12n - 44 = 0 \\ -8m + 8n - 24 = 0 \\ \end{eqnarray*}$$

  We first derive cordinates of the center $S$ and then also the radius $r$.

• Result

  The circle equation is: $(x-1)^2+(y-4)^2=10$.

  Observe that $S$ is the center of the segment $AC$, hence the triangle $ABC$ is right-angled.