The perpendicular bisectors of a triangle intersect at a point that is equidistant from the sides ofthe triangle
The statement is a well-known property of triangles called the circumcenter of a triangle
The statement is a well-known property of triangles called the circumcenter of a triangle. The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle, and it is equidistant from the three sides.
To understand why this is true, let’s consider a triangle ABC.
1. Perpendicular Bisectors: Draw the perpendicular bisectors of sides AB, BC, and AC. These bisectors are lines that pass through the midpoints of the sides and are perpendicular to those sides. Let’s call the perpendicular bisectors BD, AE, and CF, respectively.
2. Concurrency: The first thing to note is that the perpendicular bisectors are intersecting at a single point. This is because any point on the perpendicular bisector of a side is equidistant to the endpoints of that side. So, if we take any two perpendicular bisectors, their point of intersection will satisfy this equidistant condition for both sides.
3. Equidistant from Sides: Now, let’s prove that the point of intersection, which we call the circumcenter O, is equidistant from all three sides.
To prove that O is equidistant from sides AB and AC, we can draw radii OD and OE from O to sides AB and AC, respectively. Since O lies on perpendicular bisectors BD and AE, it is equidistant from A and B, as well as B and C. Therefore, we have:
OD = OB (equidistant from sides AB)
OE = OC (equidistant from sides AC)
Similarly, we can draw a radius OF from O to side BC. Since O lies on the perpendicular bisector CF, it is equidistant from B and C. Therefore:
OF = OB (equidistant from sides BC)
Combining the above results, we have:
OD = OA = OE (equidistant from sides AB and AC)
OF = OB = OC (equidistant from sides BC)
Hence, the point O, which is the intersection of the perpendicular bisectors of triangle ABC, is equidistant from all three sides. This is a characteristic property of the circumcenter of a triangle.
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The Intersection of Angle Bisectors: Proving Equal Distances from Point O to Triangle Sides