https://file.aiquickdraw.com/mathbot/file/1726977144529h6u9q21w.jpg

We are given that:

\[
\overarc{AB} \cong \overarc{BC} \cong \overarc{CA}
\]

We need to prove that \( \triangle ABC \) is equiangular.

### Step-by-Step Proof:

1. **Understanding the Given Information**:
   - The problem specifies that \( \overarc{AB} \cong \overarc{BC} \cong \overarc{CA} \), meaning the arcs subtended by the sides of the triangle \( \triangle ABC \) on the circle are congruent.
   - Since the arcs are congruent, the corresponding central angles that subtend these arcs must also be congruent.

2. **Apply the Circle Theorem (Equal Arcs Subtend Equal Angles)**:
   - The angles subtended by arcs at the center of the circle are equal if the arcs themselves are equal.
   - Therefore, the central angles \( \angle AOB = \angle BOC = \angle COA \), where \( O \) is the center of the circle, are all equal.

3. **Inscribed Angles Theorem**:
   - The inscribed angle subtended by a chord is half the central angle subtended by the same chord.
   - Since the central angles are equal, the inscribed angles at each vertex of the triangle must also be equal.
   
   So, \( \angle A = \angle B = \angle C \).

4. **Conclusion**:
   - Since all the angles in \( \triangle ABC \) are equal, we conclude that \( \triangle ABC \) is equiangular.
   
Thus, we have proved that \( \triangle ABC \) is equiangular.

---

Would you like more details on any step? Here are some related questions to deepen your understanding:

1. How does the inscribed angle theorem relate to the central angle?
2. Why does congruent arcs imply equal angles in the context of a circle?
3. What is the significance of the circle in making triangle \( \triangle ABC \) equilateral?
4. Can an equilateral triangle also be described as equiangular? Why?
5. How would you prove that the sides of \( \triangle ABC \) are also equal?

**Tip**: In any circle, congruent arcs always lead to congruent angles at the circumference. Use this concept when dealing with geometrical proofs in circles!

Given: AB ≅ BC ≅ CA. Prove: △ABC is equiangular.

Learn how to prove that triangle ABC is equiangular given equal arcs AB ≅ BC ≅ CA. This problem involves circle theorems and equilateral triangle properties, making it suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation