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In the diagram below, if the red lines are perpendicular bisectors, the point at which they cross is called the ______.

Question

In the diagram below, if the red lines are perpendicular bisectors, the point at which they cross is called the ______.

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Solution

The point at which the perpendicular bisectors of a triangle (or any other shape) cross is known as the circumcenter. The circumcenter has some interesting properties: it is equidistant from all three vertices of the triangle, which means that a circle drawn with the circumcenter as the center and a radius equal to the distance from the circumcenter to any vertex will pass through all three vertices. This property makes the circumcenter significant in various geometric constructions and proofs. Additionally, in obtuse triangles, the circumcenter lies outside the triangle, while in acute triangles, it lies inside, and for right triangles, it sits at the midpoint of the hypotenuse. Thus, the answer to the question is circumcenter.

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