## what is the uquidistant from the vertices of a triangle?

### circumcenter

The concept of being equidistant from the vertices of a triangle can be understood by considering the circumcenter of the triangle. The circumcenter is the center of the circle that passes through all three vertices of the triangle.

To find the equidistant point, draw the perpendicular bisectors of each side of the triangle. The point where these bisectors intersect is the circumcenter and is equidistant from all three vertices of the triangle.

Let’s break down the process of finding the circumcenter:

1. Draw a triangle with three vertices A, B, and C.

2. Find the midpoint of side AB and draw a perpendicular line that passes through this midpoint. Repeat this process for sides BC and AC, finding the midpoints and drawing perpendicular lines.

3. The three perpendicular bisectors will intersect at a single point, which is the circumcenter.

4. This circumcenter is equidistant from the three vertices of the triangle, meaning it is the same distance away from each vertex.

It is worth noting that not all triangles have a circumcenter. Only triangles that are not degenerate (meaning they are not collapsed to a line or a point) will have a circumcenter. Additionally, some triangles can have an incentre or orthocenter which are different from the circumcenter.

I hope this clarifies the concept of being equidistant from the vertices of a triangle! Let me know if you have any further questions.

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