Circum Centre Of Triangle

The circumcenter of a triangle is the point of intersection of the perpendicular bisectors of its sides. Here are some important points about the circumcenter of a triangle:

1. The circumcenter is the center of the circumcircle, which is the circle passing through all three vertices of the triangle.
2. The circumcenter is equidistant from the three vertices of the triangle.
3. The circumcenter is not necessarily inside the triangle. It can be outside the triangle for an obtuse triangle and on the triangle for a right triangle.
4. The circumcenter is located at the intersection point of the perpendicular bisectors of any two sides of the triangle.
5. The circumcenter of an equilateral triangle coincides with its centroid, incenter, and orthocenter.
6. The circumcenter of a right triangle is located at the midpoint of the hypotenuse.
7. The circumcenter of an isosceles triangle lies on the line of symmetry.
8. The circumcenter of a triangle can be used to find the radius of the circumcircle using the formula R = a/2sinA = b/2sinB = c/2sinC, where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the angles opposite those sides.
9. The circumcenter is a point of concurrency of the triangle.
10. The circumcenter of a triangle is unique, except in the case of an isosceles triangle where the circumcenter lies on the line of symmetry.