[tex]\displaystyle\\
\text{In }\Delta ABC \text{ avem:}\\
MN~||~AC ~\Longrightarrow~ \frac{BN}{BC} = \frac{BM}{BA} ~~\text{Conform T. Thales}\\\\
\text{In }\Delta ABD \text{ avem:}\\
MP~||~AD ~\Longrightarrow~ \frac{BM}{BA} = \frac{BP}{BD} ~~\text{Conform T. Thales}\\\\
a)\\
\text{Daca }\frac{BN}{BC} = \underbrace{\frac{BM}{BA}~\texrt{si}~\frac{BM}{BA}} = \frac{BP}{BD}~\Longrightarrow~\underbrace{\frac{BN}{BC} =\frac{BP}{BD}~\Longrightarrow~NP~||~CD}_{\texttt{conform R. T. Thales}}
[/tex]
[tex]\displaystyle\\
b)\\
\text{Daca }~\frac{BN}{BC} =\frac{BP}{BD}~\text{atunci:}\\\\
\frac{CN}{BC} +\frac{BP}{BD}=\frac{CN}{BC} +\frac{BN}{BC}=\frac{CN+BN}{BC} =\frac{BC}{BC}=1[/tex]
