More Theorems About Triangles
Rule

Circumcenter Theorem

The circumcenter of a triangle is equidistant to the vertices of the triangle.
Circumcenter S of a triangle ABC

Based on the characteristics of the diagram, the following relation holds true.

Proof

Assume that is a triangle and and are the perpendicular bisectors of the sides of this triangle.

A Triangle ABC with perpendicular bisectors DS, FS, ES

Notice that is a point on the perpendicular bisector of Therefore, by the Perpendicular Bisector Theorem, is equidistant from and

AS=BS

Similarly, is also a point on the perpendicular bisector of Using the Perpendicular Bisector Theorem once again, it can be concluded that is equidistant from and

BS=CS
By the Transitive Property of Equality, is equal to
This proves that and are all equal.
AS=BS=CS

Two-Column Proof

The proof can be summarized in the following two-column table.

Statements Reasons
Given
Perpendicular Bisector Theorem
Transitive Property of Equality