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The diagram shows an equilateral triangle \( BDF \), with three tangents \( ABC \), \( CDE \), and \( AFE \) to a circle. Since the triangle is equilateral, all sides and angles are equal.

### Key observations:
1. In an equilateral triangle, each interior angle is \( 60^\circ \).
2. The points \( A \), \( B \), and \( C \) form tangents to the circle.
3. You are asked to calculate unknown angles and values, labeled as \( x \) and \( 3y \), based on these properties.

### Analysis:
- Since the triangle is equilateral, all three angles at \( B \), \( D \), and \( F \) are \( 60^\circ \). 
- \( x \) refers to the angle inside one of the smaller triangles formed by the tangents and the points of contact with the circle.
- \( 3y \) seems to be connected to an exterior or secondary angle that involves breaking down the \( 60^\circ \) angles further. 

Let me know if you'd like a detailed solution approach for calculating \( x \) and \( 3y \). 

Here are a few related questions that could help expand understanding:
1. What are the properties of tangents to a circle from an external point?
2. How do the angles in a triangle formed by tangents to a circle behave?
3. Can you explain how the sum of angles at the tangency points would help solve for \( x \) and \( 3y \)?
4. How can symmetry in an equilateral triangle simplify geometric calculations?
5. Why does an equilateral triangle ensure equal tangency conditions on the circle?

**Tip**: Use the symmetry of the equilateral triangle to simplify angle calculations and keep in mind that tangents from a common external point are equal in length.

The diagram shows an equilateral triangle of BDF. ABC, CDE, and AFE are tangents to the circle. Find the angles x and 3y.

Explore how to calculate angles x and 3y in an equilateral triangle with tangents to a circle. Learn the application of geometric theorems involving tangents, triangles, and angle sums. Suitable for students in Grades 9-11.

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