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In order to find the midpoint of segment AB, you add together the X-Coordinates and divide them by two, then add together the Y-Coordinates and divide them by two, which will give you (-1.5, 2). In order to find the midpoint of segment AC, you add together the X-Coordinates and divide them by two, then add together the Y-Coordinates and divide them by two, which will give you (2, 1.5). Finding the midpoint of segment BC is not necessary, because you only need two perpendicular bisectors to find the circumcenter.



D is the midpoint of AB, and E is the midpoint of AC.

Then construct the perpendicular bisectors to find the circumcenter.

To construct the perpendicular bisectors, you must first find the negative reciprocal of the original slope of the line. Then, you insert the midpoint coordinates into the equation for X and Y. Solve for B, or the Y-Intercept, to complete the equation for the line.

The slope of segment AB is 6/5(found by using points A and B), which means the negative reciprocal would be -5/6. -5/6 would be the slope of the perpandicular bisector for line AB, through point D. Insert the coordinate of D, being -1.5,2, into the equation for X and Y, and you have 2=-5/6(-1.5)+b. Solving through the equation algebraicly will show you that B is 3/4. The final equation for the perpendicular bisector containing midpoint D is Y=-5/6X+3/4.

The slope of segment AC is -7/2(found by using points A and C), which means the negative reciprocal would be 2/7. 2/7 would be the slope of the perpandicular bisector for line AC, through point E. Insert the coordinate of E, being 2,1.5, into the equation for X and Y, and you have 1.5=2/7(2)+b. Solving through the equation algebraicly will show you that B is 13/14. The final equation for the perpendicular bisector containing midpoint E is Y=2/7X+13/14.

When finding the circumcenter, being that little point that we put on but didn't label with a letter,

As you can see from our long and tedious work below, the circumcenter is (-15/94,83/94). You must use a system of equations with your two equations of the perpendicular bisectors you found. First you solve for X, then put X into the original two equations to solve for Y. Thus, the circumcenter is born.

To find the distance from one of the points, as shown below, you must subtract the circumcenter from one of the points. With the new point, which happens to be (1.15957, 4.11702), you square the two coordinates, add them together, and then find the square root of that number. This will be the radius of the circle from the circumcenter to one of the vertices of the triangle. Then, by using a compass or Geometer's Sketchpad, you can trace your circle, which should run through all vertices of the triangle, as shown in our very first picture at the top of the page.