Chapter 3.8, Problem 16PSC

To determine

To prove : $\Delta DEC$ is equilateral.

Explanation of Solution

Given information : The following information has been given

  $\angle BEC\cong \angle ACB={90}^0$

  $\overrightarrow{AC}\cong \overrightarrow{BE}$

  $\overrightarrow{DE}\cong \overrightarrow{EC}$

Formula used : Information/data provided in the question that are used to solve this particular category of solution/solution.

Proof : In triangles BEA and BED, we can say that

  $\angle BEA\cong \angle BED={90}^0$

BE is the common chord

  $\overrightarrow{AB}\cong \overrightarrow{BD}$

Thus, we can say that that two triangles are congruent. Now, using virtue of congruency, we can say that

  $\overrightarrow{AE}\cong \overrightarrow{DE}$

Hence, we also know that in triangle AEC,

  $\overrightarrow{AE}\cong \overrightarrow{EC}$

Thus, from the above two equations, we can say that

  $\overrightarrow{DE}\cong \overrightarrow{EC}\cong \overrightarrow{DC}$

Hence, proved that $\Delta DEC$ is equilateral