In this chapter, we will show how to copy a triangle ${\displaystyle \triangle ABC}$ to other triangle ${\displaystyle \triangle CDE}$ . The construction is an excellent example of the reduction technique – solving a problem by solution to a previously solved problem.

The construction

1. Construct a triangle from the sides of ${\displaystyle \triangle ABC}$ : ${\displaystyle {\overline {AB}}}$, ${\displaystyle {\overline {AC}}}$, ${\displaystyle {\overline {BC}}}$ and get ${\displaystyle \triangle CDE}$.

Claim

The triangles ${\displaystyle \triangle ABC}$ and ${\displaystyle \triangle CDE}$ are congruent.

Proof

1. ${\displaystyle {\overline {AB}}}$, ${\displaystyle {\overline {AC}}}$, ${\displaystyle {\overline {BC}}}$ are sides of the triangle ${\displaystyle \triangle ABC}$ and therefore obey the triangle inequality.
2. Therefore one can build a triangle whose sides equal these segments.
3. The sides of the triangle ${\displaystyle \triangle ABC}$ and ${\displaystyle \triangle CDE}$ are equal.
4. Due to The Side-Side-Side congruence theorem the triangles ${\displaystyle \triangle ABC}$ and ${\displaystyle \triangle CDE}$ congruence.