Circumcenter Theorem
Circumcenter
The threeperpendicular bisectors一个三角形满足在一个point, called thecircumcenter.
Circumcenter Theorem
The vertices of a triangle are equidistant from the circumcenter.
Given:
, the perpendicular bisectors ofand.
To prove:
The perpendicular bisectors intersect in a point and that point is equidistant from the vertices.
The perpendicular bisectors ofandintersect at point.
Let us prove that pointlies on the perpendicular bisector ofand it is equidistant from,and.
Drawand.
Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
So,and.
By the transitive property,
.
Any point equidistant from the end points of a segment lies on its perpendicular bisector.
So,is on the perpendicular bisector of.
Since, pointis equidistant from,and.
This means that there is a circle having its center at the circumcenter and passing through all three vertices of the triangle. This circle is called thecircumcircle.