Question

Find the circumcenter of a triangle with coordinates A(0,0), B(0,6), C(4,0).
The coordinates of the circumcenter are: (?, ?)

Answer 1

There are two ways I can think of to approach the problem. Both stem from different ways to understand what a circumcenter is.

One way to think of the circumcenter is that it's the *intersection of the perpendicular bisectors of the three sides of a triangle*. From this perspective, it makes sense to try and draw the triangle. If you plot the points A, B, and C on the plane, you'll notice that the triangle ABC is a right triangle. Try plotting it. Then trying drawing in the perpendicular bisectors for the sides AB and AC, the legs of the triangle.
You'll notice that they intersect at a point on the hypotenuse of the triangle.

It's not a coincidence that this point is also the midpoint of the hypotenuse, and so this point also lies on the perpendicular bisector of the hypotenuse. This point must be the circumcenter.

Another way to understand the circumcenter is that it's the *center of the circle that circumscribes the triangle.* It might not be obvious, perhaps you might not have studied it yet, but there's a theorem in geometry that says "If a circle circumscribes a right triangle, then the hypotenuse of the triangle is the diameter of the circle." From that perspective, we see that the circumcenter is just the midpoint of the hypotenuse, just as we saw in the first solution above.