Suppose we have an equilateral triangle ABC:
Three lines of symmetry in triangle are BF , AD and CE. These lines divide triangle in two congruent parts each:
1. For line AD triangle ABD is congruent to triangle ACD by property of Triangles SAS (SIDE, ANGLE, SIDE) with AD as common side, AB = AC and angle BAD = angle CAD.
2. For line BF triangle ABF is congruent to triangle BCF by property of triangles SAS (SIDE, ANGLE, SIDE) with BF as common side, AB = BC and angle ABF = angle CBF.
3. For line CE triangle BCE is congruent to triangle ACE by property of triangles SAS (SIDE, ANGLE, SIDE) with CE as common side, AC = BC and angle ACF = angle BCF.
Three lines we drew AD, BF and CE are considered to be perpendicular bisectors at points D, E and F respectively of three sides of triangle ABC i.e.
BD = DC,
AF = CF and,
AE = BE.