Question

# Write a proof for the following \overline{AB}\cong \overline{DE} and C is the midpoint of both \overline{AB}\ and\ \overline{BE}

Similarity
Write a proof for the following:
Given $$\displaystyle\overline{{{A}{B}}}\stackrel{\sim}{=}\overline{{{D}{E}}}$$ and C is the midpoint of both $$\displaystyle\overline{{{A}{B}}}\ {\quad\text{and}\quad}\ \overline{{{B}{E}}}$$.
Prove: $$\displaystyle\triangle{A}{B}{C}\stackrel{\sim}{=}\triangle{D}{E}{C}$$.

2021-08-10
Step 1
$$\displaystyle{A}{B}\stackrel{\sim}{=}{D}{E}$$ and C is the mid point of both AD and BE.
Now
AC=DC...(1) ($$\displaystyle\because$$ C is mid point of AD)
CE=BC...(2) ($$\displaystyle\because$$ C is the mid point of BE)
$$\displaystyle\angle{A}{C}{B}=\angle{B}{C}{E}$$...(3) ($$\displaystyle\because$$ vertically opposite $$\displaystyle\angle{s}$$)
Also $$\displaystyle{A}{B}\stackrel{\sim}{=}{D}{E}$$...(4) (Given)
Step 2
Therefore from (1), (2) and (4)
$$\displaystyle\triangle{A}{B}{C}\stackrel{\sim}{=}\triangle{D}{E}{C}$$ (by SSS similarity)
Also from (1), (2) and (4)
$$\displaystyle\triangle{A}{B}{C}\stackrel{\sim}{=}\triangle{D}{E}{C}$$ (by SAS similarity)