Also see "Perpendicular Bisectors in a Triangle" at Segments in Triangles.

 A perpendicular bisector of a given line segment is a line (or segment or ray) which is perpendicular to the given segment and intersects the given segment at its midpoint (thus "bisecting" the segment).

 The perpendicular bisector of a line segment is the set of all points that are equidistant from its endpoints. To be discussed further in the section on Constructions.
 Every point on the perpendicular bisector, , is the same distance from point A as it is from point B. AC = CB AF = FB AG = GB AE = EB AH = HB AJ = JB AD = DB You can think of the points on the perpendicular bisector as being the third vertices of a series of isosceles triangles with vertices A and B.

 Theorem Proof: (transformational method)
 1. A perpendicular bisector of a segment (by definition) is a line that is perpendicular to the segment and intersects the segment at its midpoint. 2. because a midpoint of a segment divides the segment into two congruent segments. 3. AD = DB because congruent segments are segments of equal measure. 4. The line of reflection for a segment is the perpendicular bisector of the segment. 5. Under a reflection in , A is mapped onto B, C is mapped onto C, and D is mapped onto D. 6. Under a reflection in , is mapped onto . 7. CA = CB since a reflection is a rigid transformation which preserves length.

 Theorem Proof: (two-column method)
 Statements Reasons 1. 1. Given 2. 2. Segment bisector forms 2 congruent segments. 3. ∠ADC, ∠BDC are right angles 3. Perpendiculars form right angles. 4. ∠ADC ∠BDC 4. All right angles are congruent. 5. 5. Reflexive property 6. 6. SAS: If 2 sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 7. 7. CPCTC: Corresponding parts of congruent triangles are congruent. 8. CA = CB 8. Congruent segments have equal measure.