There are many ways to show two segments are of equal length. Let us first discuss such problems. There are too many theorems and results which can be used, but let us just list a few more common techniques:
Using congruent triangles;
Using a third line segment;
Using isosceles triangles;
Circle Properties: Chords of equal length are of the same distance from the center of the circle, chords subtended by the same angle at the center of the circle are of the same length, etc;
If a line parallel to one side of a triangle bisects a second side, it also bisects the third side;
Comparing the lengths with multiples or , ... of another length.
. Construct squares and externally to the sides and respective of triangle . Let be the foot of the perpendicular from to , and let intersect at . Show that .
. In , . Let be a point on and be a point on extended such that . Let intersect at . Show that .
. Let and be the midpoints on the sides and respectively of a parallelogram . Show that and trisect .
. Let be a triangle. Construct equilateral triangles and externally to sides and respectively. Let be the point such that is a parallelogram. Show that is equilateral.
. Let and be the altitudes of a triangle . Let be the midpoint of and be on such that . Show that .