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.