The "middle" of a triangle, Method #2:The circumcenter

Yesterday, we discussed the centroid, the simplest way to measure the center of a triangle. For a triangle drawn as three vertices and the lines that connect them, draw the lines from each vertex to the midpoint of the opposite line. They will meet in a single point and that point is the centroid.  Another way to do it is if you have the coordinates of the three points, the centroid's x value is the average of the x values and likewise its y value is the average if the y values.

Here is a different way to get a center of a triangle, the point that is the same distance away from each vertex.  The geometric solution is to find the midpoint of each line segment of the triangle then draw the perpendicular bisector of the segment. The three perpendicular bisectors will all meet at a single point. The red circle in this picture is the only circle that goes through all three points A, B and C, and as this picture shows, it's not really necessary to do all three perpendicular bisectors, because the third will also pass through the point indicated by the green arrow.

If we had the coordinates for A, B and C, we would use them to find the midpoints and slopes of each line segments, call them mid1, mid2 and mid3 and slope1, slope2 and slope3.  It's possible that one of the slopes is zero, but it isn't possible to have two slopes equal to each other, because that would mean two lines are parallel, impossible if they are sides of a triangle.  Choose two lines with non-zero slopes and take the slope of the perpendicular, which equals -1/slope.  Without loss of generality, let's assume that lines 1 and 2 don't have a zero slope.  Then we just need to solve for x and y in the following pair of simultaneous linear equations.

y - y_from_mid1 = -1/slope1(x - x_from_mid1)
y - y_from_mid2 = -1/slope2(x - x_from_mid2)

It might look daunting, but the methods are fairly straightforward.

One "unusual" attribute of a circumcenter is that it doesn't have to be on the inside of the triangle. In fact, the rules for the position are based on the classification of the triangle.

Tomorrow, classifications and circumcenters.