The constructed circumcenter is attached and labelled as P.
What is a circumcenter?
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. In other words, it is the center of the circle that circumscribes (passes through the vertices of) the triangle. The circumcenter is equidistant from the three vertices of the triangle.
To determine the circumcenter, follow these:
1. Find the midpoint of each side of the triangle.
2. Determine the slope of each side.
3. Find the negative reciprocal of each slope to get the slopes of the perpendicular bisectors.
4. Use the midpoint and slope to write the equation of each perpendicular bisector.
5. Solve the system of equations formed by the equations of the perpendicular bisectors to find the circumcenter.
The circumcenter has some interesting geometric properties. It is always inside or on the triangle for any type of triangle (acute, obtuse, or right-angled). In an equilateral triangle, the circumcenter is the centroid (the point of concurrency of the medians) and the orthocenter (the point of concurrency of the altitudes).