How to show two lengths are equal

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;

• Using parallelograms;

• 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 $\frac{1}{2}$, $\frac{1}{3}$ ... of another length.

Exercises

$1$. Construct squares $ABEF$ and $ACGH$ externally to the sides $AB$ and $AC$ respective of triangle $ABC$. Let $D$ be the foot of the perpendicular from $A$ to $BC$, and let $AD$ intersect $FH$ at $M$. Show that $FM=MH$.

$2$. In $\triangle{ABC}$, $AB=AC$. Let $D$ be a point on $AB$ and $E$ be a point on $AC$ extended such that $BD=CE$. Let $DE$ intersect $BC$ at $F$. Show that $DF=FE$.

$3$. Let $E$ and $F$ be the midpoints on the sides $BC$ and $AD$ respectively of a parallelogram $ABCD$. Show that $BF$ and $DE$ trisect $AC$.

$4$. Let $ABC$ be a triangle. Construct equilateral triangles $ABD$ and $ACE$ externally to sides $AB$ and $AC$ respectively. Let $F$ be the point such that $ADFE$ is a parallelogram. Show that $\triangle{FBC}$ is equilateral.

$5$. Let $BD$ and $CE$ be the altitudes of a triangle $ABC$. Let $F$ be the midpoint of $BC$ and $G$ be on $DE$ such that $DE \perp FG$. Show that $DG=GE$.