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It is given that AB=AD=AE, Angle ABC equals Angle ADC, and that D is a point on CE. Prove that triangles ABC and AEC have two sides and an angle in front of the shortest side which are respectively equal, but the triangles aren't congruent.

To solve this problem, let's first summarize the given information and then proceed with the proof.

### Given:
1. \( AB = AD = AE \)
2. \( \angle ABC = \angle ADC \)
3. \( D \) is a point on \( CE \).

### To Prove:
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.

### Proof:

1. **Identifying the sides and angles:**

   - Since \( AB = AD = AE \), triangles \( \triangle ABD \) and \( \triangle ADE \) are isosceles.
   - The angles \( \angle ABC \) and \( \angle ADC \) are given to be equal.
   - \( D \) lies on \( CE \), making \( AD \) a common side between \( \triangle ADC \) and \( \triangle AEC \).

2. **Compare triangles \( \triangle ABC \) and \( \triangle AEC \):**

   - **Side \( AB = AE \):** Given directly as \( AB = AE \).
   - **Angle \( \angle ABC = \angle ADC \):** This is given.
   - **Side \( BC = EC \):** Both these sides lie on the same line \( CE \) but this alone doesn’t ensure congruency or inequality.

3. **Analyze the congruency condition:**

   - Although \( \triangle ABC \) and \( \triangle AEC \) have two sides equal (\( AB = AE \) and \( BC = EC \)) and an angle (\( \angle ABC = \angle ADC \)), they are not congruent because:
     - The corresponding sides opposite the equal angles are not the same length (\( AC \neq AC \) because \( D \) divides \( CE \) into two segments).
     - \( AB \) is not necessarily equal to \( EC \) and \( AE \) is not equal to \( BC \).

4. **Conclusion:**

   The triangles \( \triangle ABC \) and \( \triangle AEC \) have two sides and the angle between them equal, but since the third sides (\( AC \) for both) do not match in length, the triangles are not congruent.

Would you like to explore further details or have any specific questions? Here are five related questions you might find interesting:

1. How does the position of point \( D \) affect the overall geometry of the triangles?
2. What is the significance of having equal angles but non-congruent triangles in geometric constructions?
3. Can you identify other pairs of triangles in this diagram that might exhibit similar properties?
4. How can we use coordinate geometry to analyze and prove this situation?
5. What conditions must change for \( \triangle ABC \) and \( \triangle AEC \) to become congruent?

**Tip:** When analyzing geometric proofs, carefully examine the implications of each given fact, particularly when it involves triangle congruency or similarity.

Explore how triangles ABC and AEC share equal sides and angles but are not congruent. Understand the geometric implications of equal angles and side lengths in triangle properties.

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