Squares ABDE and ACFH are drawn externally on the sides AB and AC respectively of a scalene `DeltaABC`. Which one of the following is correct?

Options

A. BH=CE
B. AD = AF
C. BF=CD
D. DF = EH

Answer

To solve the problem, we need to analyze the given information about the scalene triangle ABC and the squares ABDE and ACFH constructed on its sides AB and AC respectively. We will follow a systematic approach to derive the necessary relationships and congruencies.

### Step-by-Step Solution:

1. **Draw the Triangle and Squares**:
- Begin by sketching triangle ABC, ensuring it is scalene (all sides of different lengths).
- Draw square ABDE on side AB and square ACFH on side AC.

2. **Label the Points**:
- Label the vertices of triangle ABC as A, B, and C.
- For square ABDE, label the vertices as follows: A at the top left, B at the bottom left, D at the bottom right, and E at the top right.
- For square ACFH, label the vertices as follows: A at the top left, C at the bottom left, F at the bottom right, and H at the top right.

3. **Identify Angles**:
- Note that in square ABDE, angle BAE is 90 degrees, and in square ACFH, angle CAH is also 90 degrees.
- Therefore, we can write the following relationships:
- Angle BAE + Angle BAC + Angle CAE = 90° + Angle BAC + Angle CAE = 90° + Angle BAC + Angle CAH.

4. **Establish Relationships Between Angles**:
- From the previous step, we can conclude:
- Angle CAE = Angle BAH (since both angles are equal to the angle BAC).

5. **Consider Triangles EAC and HAB**:
- Now, we will analyze triangles EAC and HAB.
- In triangle EAC:
- EA = AB (sides of square ABDE)
- AC = AH (sides of square ACFH)
- Angle CAE = Angle BAH (as established earlier).

6. **Apply the SAS Congruence Criterion**:
- Since we have two sides and the included angle equal in both triangles:
- Triangle EAC is congruent to triangle HAB by the Side-Angle-Side (SAS) criterion.

7. **Conclude with Corresponding Parts**:
- By the Corresponding Parts of Congruent Triangles (CPCT), we can conclude:
- EC = BH.

### Final Result:
Thus, we have established that the segments EC and BH are equal, which is a significant relationship derived from the properties of the squares and the triangle.