Proof

Proof of Theorem 5-6

Given: Lines l, m, and n are the perpendicular bisectors of the sides of Δ A B C . cap delta eh b c .  P is the intersection of lines l and m.

Prove: Line n contains point P, and P A = P B = P C . p eh equals . p b equals p c .

Proof: A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Point P is on l, which is the perpendicular bisector of A B ¯ , eh b bar , comma  so P A = P B . p eh equals p b .  Using the same reasoning, since P is on m, and m is the perpendicular bisector of B C ¯ , P B = P C . b c bar , comma . p b equals p c .  Thus, P A = P C p eh equals p c  by the Transitive Property. Since P A = P C , p eh equals p c comma  P is equidistant from the endpoints of A C ¯ . eh c bar , .  Then, by the converse of the Perpendicular Bisector Theorem, P is on line n, the perpendicular bisector of A C ¯ . eh c bar , .

Problem 1 Finding the Circumcenter of a Triangle

What are the coordinates of the circumcenter of the triangle with vertices P(0, 6), O(0, 0), and S(4, 0)?

Find the intersection point of two of the triangle's perpendicular bisectors. Here, it is easiest to find the perpendicular bisectors of P O ¯ p o bar  and O S ¯ . o s bar , .

Image Long Description

The coordinates of the circumcenter of the triangle are (2, 3).

✓ Got It?

1. What are the coordinates of the circumcenter of the triangle with vertices A(2, 7), B(10, 7), and C(10, 3)?