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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 (xm)2+(yn)2=r2. Subtract the quadratic terms to get the coordinates of the centre S=(m,n).

  • Resolution

    We get equations

    (2m)2+(1n)2=r2(4m)2+(3n)2=r2(0m)2+(7n)2=r2

    After the expansion

    4m+m22n+n2+5=r28m+m26n+n2+25=r2m214n+n2+49=r2

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

    4m+12n44=08m+8n24=0

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

  • Result

    The circle equation is: (x1)2+(y4)2=10.

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

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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