Triangle epicenter theorem

The center of the circumscribed circle of a triangle is called the outer center of the triangle.

The nature of the external world:

1. The perpendicular bisector of three sides of a triangle intersect at a point, which is the outer center of the triangle.

2. If O is the outer center of △ABC, ∠BOC=2∠A(∠A is acute angle or right angle) or ∠ BOC = 360-2 ∠ A (∠ A is obtuse angle).

3. When the triangle is an acute triangle, the outer center is inside the triangle; When the triangle is an obtuse triangle, the outer center is outside the triangle; When the triangle is a right triangle, the outer center is on the hypotenuse and coincides with the midpoint of the hypotenuse.

4. To calculate the coordinates of the epicenter, we must first calculate the following temporary variables: d 1, d2 and d3 are the point multiplication of the vectors whose three vertices are connected with the other two vertices. c 1=d2d3，c2=d 1d3，C3 = d 1 D2； C=c 1+c2+c3. Coordinate of gravity center: (

(c2+c3)/2c，(c 1+c3)/2c，(c 1+c2)/2c

)。

5. The distances from the outer center to the three vertices are equal.