设P到顶点B的距离为x,∠ABP=a,∠CBP=b
sina=1/x cosa=√(x^2-1)/x
sinb=2/x cosb=√(x^2-4)/x
sin∠ABC=sin(a+b)=sin60°=√3/2
sinacosb+cosasinb=√3/2
1/x*√(x^2-4)/x+√(x^2-1)/x*2/x=√3/2
2√(x^2-4)+4√(x^2-1)=√3x^2
4x^2-16+16x^2-16+16√(x^2-1)(x^2-4)=3x^4
16√(x^2-1)(x^2-4)=3x^4-20x^2+32
256x^4-1280x^2+1024=9x^8+400x^4+1024-120x^6+192x^4-1280x^2
9x^8-120x^6+336x^4=0
3x^4-40x^2+112=0
△=256
x^2=(40±16)/6=28/3或4
x=2/3*√21或2（舍去）
所以P到顶点B的距离为2/3*√21