Circle interpolation
Task number: 2355
Interpolate a circle through points \(A=(2{,}1)\), \(B=(4{,}3)\) and \(C=(0{,}7)\).
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.
