https://file.aiquickdraw.com/mathbot/file/1724439255182j0tts3vp.jpg

Let's solve problem **3.42**:

### Problem Statement:
Triangle \(ABC\) is equilateral, and \(ABDE\), \(BCFG\), and \(CAHI\) are squares. Prove that triangle \(DFH\) is equilateral.

### Solution Outline:

1. **Establishing properties of the squares and equilateral triangle:**
   - Since \(ABC\) is equilateral, all sides are equal: \(AB = BC = CA\).
   - All angles of triangle \(ABC\) are \(60^\circ\).
   - Each square has right angles and equal sides.

2. **Finding relationships between the points \(D\), \(F\), and \(H\):**
   - \(DE = AB\), \(FG = BC\), and \(HI = CA\), since these are sides of the squares.
   - The lengths \(DF\), \(FH\), and \(HD\) should be equal since they connect corresponding vertices of these squares.

3. **Using symmetry and rotations:**
   - Square \(ABDE\) rotated by \(90^\circ\) around \(B\) places \(D\) along the perpendicular to \(AB\) through \(B\).
   - Similarly, the square \(BCFG\) rotated by \(90^\circ\) around \(C\) places \(F\) along the perpendicular to \(BC\) through \(C\).
   - The square \(CAHI\) rotated by \(90^\circ\) around \(A\) places \(H\) along the perpendicular to \(CA\) through \(A\).

4. **Angle Calculations:**
   - Because triangle \(ABC\) is equilateral and each rotation by \(90^\circ\) preserves the angle between these points, the angles formed at \(D\), \(F\), and \(H\) should also be \(60^\circ\).
   - Hence, \(\angle DFH = 60^\circ\), \(\angle FHD = 60^\circ\), and \(\angle HDF = 60^\circ\).

5. **Conclusion:**
   - Since \(DFH\) has all sides equal and all angles \(60^\circ\), triangle \(DFH\) is equilateral.

This completes the proof that triangle \(DFH\) is equilateral.

Would you like a more detailed explanation of any part, or do you have any questions?

---

**Related Questions:**
1. How does the property of equilateral triangles help in proving other geometric figures equilateral?
2. What role does rotational symmetry play in geometric proofs like this?
3. How can the Pythagorean theorem be applied in problems involving squares and triangles?
4. What are some other geometric properties that can be proven using square constructions?
5. How can vector geometry be used to prove that triangle \(DFH\) is equilateral?

**Tip:** 
In problems involving symmetry, consider how rotations and reflections preserve distances and angles. This can simplify the proof significantly.

Explore the proof that triangle DFH is equilateral using squares constructed on sides of equilateral triangle ABC. Detailed explanation includes rotational symmetry and properties of equilateral triangles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation