Chapter 12.4, Problem 41E

To determine

To Prove: $\Delta DEF$ is an equilateral triangle.

Explanation of Solution

Given: $\Delta ABC$ is an equilateral triangle.

  

Here

  $\begin{array}{l}\angle ACB=\angle ABC=\angle BAC\\ AB=BC=CA\\ and\\ \angle CAF=\angle ABE=\angle BCF\\ Thus\\ \angle BAD=\angle BAC-\angle CAF\\ \angle ACF=\angle ACB-\angle BCF\\ \angle CBE=\angle ABC-\angle ABE\\ and\\ \angle BAD=\angle ACF=\angle CBE\\ Similarly\\ \angle EDF=\angle DEF=\angle EFD\end{array}$

Thus

  $\Delta DEF$ is an equilateral triangle.

Hence proved.