Task 539
Task 539: development task in emrekuruu/math-openenv, an RL environment on the Hugging Face Hub. An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 2 \angle D$ and $\angle BAC = k…
The task
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 2 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$. [asy] import graph; unitsize(2 cm); pair O, A, B, C, D; O = (0,0); A = dir(90); B = dir(-30); C = dir(210); D = extension(B, B + rotate(90)*(B), C, C + rotate(90)*(C)); draw(Circle(O,1)); draw(A--B--C--cycle); draw(B--D--C); label("$A$", A, N); label("$B$", B, SE); label("$C$", C, SW); label("$D$", D, S); [/asy]