赛昆数学挺棒。 可以稍微简单一点,不用反正切函数
tan(∠OCB)=r/x
记 y = x/3 = EF, tan(∠OAB)= tan(∠OCB) = OF/(2y)
OF/y = tan(∠OBA) = tan(90-2∠OCB) = cot(2∠OCB)
于是 2 tan(∠OCB) = cot(2∠OCB) = (1-tan^2(∠OCB))/(2 tan(∠OCB)). 解出 tan^2(∠OCB) = 1/5.
x = r/tan(∠OCB) = √(5) r
tan(∠OCB)=r/x
记 y = x/3 = EF, tan(∠OAB)= tan(∠OCB) = OF/(2y)
OF/y = tan(∠OBA) = tan(90-2∠OCB) = cot(2∠OCB)
于是 2 tan(∠OCB) = cot(2∠OCB) = (1-tan^2(∠OCB))/(2 tan(∠OCB)). 解出 tan^2(∠OCB) = 1/5.
x = r/tan(∠OCB) = √(5) r