{"status": "success", "data": {"description_md": "Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$? \n\n
\n\n

\n\n$\\textbf{(A) } \\frac{1}{14} \\qquad \\textbf{(B) } \\frac{1}{12} \\qquad \\textbf{(C) } \\frac{1}{10} \\qquad \\textbf{(D) } \\frac{3}{28} \\qquad \\textbf{(E) } \\frac{1}{9}$", "description_html": "

Circle C_1 and C_2 each have radius 1, and the distance between their centers is \\frac{1}{2}. Circle C_3 is the largest circle internally tangent to both C_1 and C_2. Circle C_4 is internally tangent to both C_1 and C_2 and externally tangent to C_3. What is the radius of C_4?


\\textbf{(A) } \\frac{1}{14} \\qquad \\textbf{(B) } \\frac{1}{12} \\qquad \\textbf{(C) } \\frac{1}{10} \\qquad \\textbf{(D) } \\frac{3}{28} \\qquad \\textbf{(E) } \\frac{1}{9}

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2023 AMC 10A Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc10A_p23", "prev": "/problem/23_amc10A_p21"}}