Math CentralQuandaries & Queries


Question from Glen, a student:

I'm having trouble finding the circumcentre of these 3 co-ordinates- A(1,1) B(10.5,1) and C(6,6).
I understand what circumcentre is but I cannot work out where it is.

With any problem like this, the first step is to draw an accurate picture. You have to find the perpendicular bisectors of two of the sides and find the point where those two lines meet. It should be obvious which two sides of the triangle you have been given are the easiest to work with. You then must

  1. find the midpoints of the two sides you have chosen;

  2. find the slopes of the lines through those points that are perpendicular to the sides;

  3. write the equations of those two lines;

  4. find the point of intersection by solving the resulting pair of equations.

For the three points you have been given, these four tasks are VERY simple. In fact, the picture will give you the answer immediately. The simple algebra should just confirm what your eyes tell you. Of course, there is a formula that gives the coordinates of the circumcentre when you plug in the coordinates of the three given points; you can find it on the internet. In this problem, however, it is far simpler not to use that formula.


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