## Circle interpolation

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.