Understanding the Circumcenter in a Triangle: Properties and Applications

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the sides ofthe triangle

The statement is true

The statement is true. In a triangle, the perpendicular bisectors of the sides intersect at a single point called the circumcenter. One of the properties of the circumcenter is that it is equidistant from the three sides of the triangle.

To understand why this is true, let’s consider a triangle ABC. The perpendicular bisector of side AB is a line that is perpendicular to AB and passes through the midpoint of AB. Similarly, the perpendicular bisector of side AC is a line that is perpendicular to AC and passes through the midpoint of AC. These two perpendicular bisectors will intersect at a single point, which we call the circumcenter.

Now, let’s examine the distances from the circumcenter to each of the sides of the triangle. Since the circumcenter is on the perpendicular bisector of AB, it will be equidistant from points A and B. This means that the distance from the circumcenter to side AB is the same as the distance from the circumcenter to side BC. Similarly, the distance from the circumcenter to side AC is also the same. Therefore, the circumcenter is equidistant from all three sides of the triangle.

This property of the circumcenter being equidistant from the sides of the triangle is useful in many geometrical proofs and constructions. It is also used to find the circumcenter in practice by constructing the perpendicular bisectors of the sides of a given triangle.

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