Concurrency of Perpendicular Bisectors
I want to prove that the perpendicular bisectors of any triangle are concurrent.
Letbe any triangle, let D and E be the midpoints of AB and BC, respectively and let l and m be the perpendicular bisectors of AB and BC, respectively. Let P be the point of intersection of l and m. The triangle is pictured below.
Now, make segments connecting P to each of the vertices of the triangle.
Notice that four new right triangles are formed (, , , and ). First look at and . Since D is the midpoint of AB, AD = BD and since l is the perpendicular bisector of AB, and : so . By reflexivity . So by SAS, . By definition of congruent triangles, . Similarly, and. Since and , by transference, .
Now look at . Since , . Let F be the midpoint of AC and make a segment between P and F.
By construction, . Since , , and , we have SAS, so . By definition of congruent triangles, . Since can be extended to be a line, . So if , we find that: