Key takeaway: The circumcenter is the meeting point of the three perpendicular bisectors of a triangle, and it serves as the center of the one circle running through all three corners.
Grade 10 – GeometryG-C.A.3Geometry and Measurement
Definition of Circumcenter
The circumcenter of a triangle is the one point that is equally far from all three vertices. Each perpendicular bisector holds the points that are equally far from two corners. Where two of them cross, the distances to all three corners agree, and the third line must pass through that spot too.
That common distance is the circumradius \(R\). A circle with that center and radius \(R\) passes through every corner. For a sharp-cornered triangle the point is inside, for an obtuse one it lands outside, and for a right triangle it sits on the longest side.
Formula and notation
For a triangle \(ABC\) with circumcenter \(O\):
\[ OA = OB = OC = R \]
For a right triangle, \(O\) is the midpoint of the hypotenuse, so \(R\) equals half the hypotenuse.
Key facts
- It is found by crossing the perpendicular bisectors of any two sides.
- Where it lands depends on the triangle: inside, outside, or on the longest side.
- The circle drawn around the triangle touches all three vertices.
- Do not confuse this point with the incenter, which comes from angle bisectors.
Circumcenter example
A sprinkler must reach three trees planted at the points \(P\) at \((0, 0)\), \(Q\) at \((10, 0)\) and \(R\) at \((0, 24)\), on a grid of yards. Where should it stand to be equally far from all three, and what is that distance?
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The corner at \(P\) is square, so side \(QR\) is the hypotenuse, and the sprinkler spot is the middle of \(QR\).
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Middle of \(QR\): \((\frac{10 + 0}{2},\ \frac{0 + 24}{2}) = (5, 12)\).
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Distance to \(P\): \(\sqrt{25 + 144} = 13\) yards, and it matches the distance to \(Q\) and \(R\).
The sprinkler stands at \((5, 12)\), 13 yards from each tree.
Common mistake
Watch out
Gus locates the sprinkler spot by drawing the medians of the triangle and taking their meeting point.
Why it is wrong
The medians meet at the centroid, a different point. Equal distance from the corners requires perpendicular bisectors, lines that cross each side at its midpoint at a right angle. Only for an equilateral triangle do the two points coincide.
Related terms and lessons
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