The problem shows a triangle that is tangent to a circle at three points. We are tasked with finding the values of $x$ and $y$. The sides of the triangle are labeled with expressions involving $x$ and $y$:

• One side has length $22$,
• Another side is $6x + 2y$,
• The third side is $4x - 2y$,
• The smaller segments of the sides near the circle’s tangent points are labeled as $x$.

Solution Concept

In problems where a triangle is inscribed around a circle, the sum of the two tangents from a common external point to the circle is equal. This means:

1. The sum of the two tangents from the left vertex to the points of tangency (both labeled $x$) should equal the length of the opposite side: $22 = (6x + 2y) + (4x - 2y)$

2. Additionally, the side lengths of the triangle must satisfy certain relations based on the geometry of the tangents.

I will now proceed to solve the system of equations to find the values of $x$ and $y$.From the equation, we have $x = \frac{11}{5}$.

Now, to fully solve for $y$, we can substitute $x = \frac{11}{5}$ into the expressions for the triangle's sides and use the condition that the triangle's perimeter should be equal to the sum of these three sides.

Let me proceed with that.It appears that there was no solution using the perimeter condition alone. Let me revise the approach by directly solving for $y$ using other available constraints or double-checking the relations of tangents and side lengths.

I'll explore further geometric principles related to this configuration to find a suitable equation for $y$.To determine the correct approach for finding $y$, let's focus on the fact that for a triangle inscribed around a circle (tangential triangle), the sum of the tangents from each vertex is equal. We can set up another equation involving $y$ based on the expressions for the sides and tangents.

Let me review the relations and refine the setup. I’ll consider the perimeter relations and reanalyze the configuration.

Would you like me to continue using this approach or explore another method for solving $y$? Feel free to ask for any clarifications!