三角形ABC中,向量AB*AC=9,sinB=cosAsinC,面积为6,p为线段

三角形ABC中,A,B,C对的边为a,b,c
sinB=cosAsinC==>sin(A+C)=cosAsinC
==>sinAcosC+cosAsinC=cosAsinC
==>sinAcosC=0==>C=π/2
AB*AC=9==>cbcosA=9 (1)
SΔ=6==>1/2 cbsinA=1/2ab=6 (2)
SΔ=1/2ab=6==>ab=12
(2)/(1):tanA=4/3,sinA=4/5,cosA=3/5
bc=15,c^2=a^2+b^2,
==>c=5,b=3,a=4
向量CP=X*向量CA/|CA|+y*向量CB/|CB|
=X/b *向量CA+y/a*向量CB
∵A,P,B三点共线
∴x/b+y/a=1
1/x+1/y=(1/x+1/y)(x/b+y/a)
=1/3+1/4+x/(3y)+y/(4x)
≥1/3+1/4+2√(1/12)=7/12+√3/3
当x/(3y)=y/(4x)时取等号
∴1/x+1/y的最小值为7/12+√3/3