Math CentralQuandaries & Queries


Question from Randy, a student:

Hi! How do I determine the equation of a circle when it is circumscribed by a triangle whose vertices are (-1, 6), (3, -2), and (2, 5)?

With lots of thanks, Randy

Hi Randy.

Key idea: The perpendicular bisector of a chord of a circle will pass through the center of the circle.

Each of the sides of the triangle is a chord of the circle. Choose a side and calculate the slope and midpoint of the side. This will allow you to write the equation of the perpendicular bisector of the side.

When you do this for two different sides, you have two lines that must intersect in the center of the circle.

Use normal methods of finding the intersection of two lines (i.e. substitution or elimination method of solving simultaneous equations) to find the center of the circle.

Once you know the center of the circle, use the distance formula between the center and any vertex of the triangle to determine its radius.

Finally, plug the center and radius into the standard form equation of a circle.

Stephen La Rocque.

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