Estude! Com Prof. Caju

Na figura se CN = a, NH = b e HQ = c. Calcular NQ
Se F, N, Q e P são pontos de tangência.
A) [tex3]\frac{a}{a+c}[/tex3]
B) [tex3]\frac{b}{a+b}[/tex3]
C) c[tex3]\sqrt{\frac{b}{a+b}}[/tex3]
D) c[tex3]\sqrt{a+b}[/tex3]
E) c[tex3]\sqrt{\frac{a}{a+b}}[/tex3]

[tex3]\mathsf{
\triangle CFN \sim \triangle CHA \implies CF:CN=CH:CA\\
CQ^2=CA⋅CF=CN⋅CH=a(a+b)\\
Seja~ γ=∠HCQ\\
T. cossenos\triangle CHQ: HQ^2 =CH^2+CQ^2-2CH.CQcosγ \implies c^2 = (a+b)^2+a(a+b)-2CQcos γ\\
2CQcosγ=\frac{-c^2+2a^2+3ab+b^2}{a+b}\\
T.cossenos\triangle CNQ: NQ^2=a(a+b)+a^2−a⋅\underbrace{2CQcosγ}.\\
Substituindo: NQ^2=\frac{ac^2}{a+b}}\\ \therefore \boxed{NQ =c.\sqrt{\frac{a}{a+b} }
}[/tex3]
(Solução:intelligentipauca)