To prove that triangles $\triangle ABC$ and $\triangle AEC$ have two sides and an angle in front of the shortest side which are respectively equal but the triangles aren't congruent, let's proceed with the following steps:

Given Information:

1. $AB = AD = AE$
2. $\angle ABC = \angle ADC$
3. $D$ is a point on $CE$.

Proof:

1. Identify the two triangles:

   • We are working with $\triangle ABC$ and $\triangle AEC$.

2. Check for equal sides:

   • $AB = AE$ (given)
   • $AC$ is common to both triangles $\triangle ABC$ and $\triangle AEC$.

3. Check for equal angles:

   • $\angle ABC = \angle ADC$ (given)
   • Notice that $\angle ABC$ is part of $\triangle ABC$ and $\angle ADC$ is part of $\triangle AEC$.

4. Side-Angle-Side (SAS) similarity:

   • For triangles to have two sides and the included angle equal, the triangles might be similar by SAS (Side-Angle-Side) similarity.
   • In $\triangle ABC$ and $\triangle AEC$:
     • $AB = AE$ (equal sides)
     • $AC$ is common (equal sides)
     • $\angle BAC$ (included angle) is common in both triangles $\triangle ABC$ and $\triangle AEC$.

   Thus, $\triangle ABC$ and $\triangle AEC$ have two sides equal and the angle between them equal.

5. Non-congruence:

   • The triangles are not congruent because $BC$ is not equal to $EC$. In $\triangle ABC$, $BC$ is a side, and in $\triangle AEC$, $EC$ is a side.
   • Since $BC$ and $EC$ are different, the two triangles cannot be congruent.

Conclusion:

• Triangles $\triangle ABC$ and $\triangle AEC$ have two sides and the included angle respectively equal. However, they are not congruent because their third sides $BC$ and $EC$ are not equal.

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. What is the significance of the angle equality $\angle ABC = \angle ADC$ in proving similarity or congruence?
2. Can we prove that triangles $\triangle ABC$ and $\triangle AEC$ are similar, and if so, under what criteria?
3. How would the proof change if $AB$ were not equal to $AE$?
4. What role does the point $D$ on $CE$ play in this geometric configuration?
5. What other geometric properties can be deduced from this configuration?

Tip: When dealing with triangles, remember that congruence implies similarity, but similarity does not necessarily imply congruence.