Problem
1964 AHSME Problem 36
In this figure the radius of the circle is equal to the altitude of the equilateral triangle ABC. The circle is made to roll along the side AB, remaining tangent to it at a variable point T and intersecting lines AC and BC in variable points M and N, respectively. Let n be the number of degrees in arc MTN. Then n, for all permissible positions of the circle:
\begin{array}{l} \textbf{(A) }\text{varies from }30^{\circ}\text{ to }90^{\circ}\quad \textbf{(B) }\text{varies from }30^{\circ}\text{ to }60^{\circ} \\ \textbf{(C) }\text{varies from }60^{\circ}\text{ to }90^{\circ} \quad \textbf{(D) }\text{remains constant at }30^{\circ}\quad \textbf{(E) }\text{remains constant at }60^{\circ} \end{array}
![[asy] pair A = (0,0), B = (1,0), C = dir(60), T = (2/3,0); pair M = intersectionpoint(A--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)), N = intersectionpoint(B--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)); draw((0,0)--(1,0)--dir(60)--cycle); draw(Circle((2/3,sqrt(3)/2),sqrt(3)/2)); label("$A$",A,dir(210)); label("$B$",B,dir(-30)); label("$C$",C,dir(90)); label("$M$",M,dir(190)); label("$N$",N,dir(75)); label("$T$",T,dir(-90)); //Credit to bobthesmartypants for the diagram[/asy]](/static/aops/b106a815964ddbe0ea5fe4de13ddb6b2ab5de29d.png)
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