Question

In dreptunghiul ABCD , se considera M un punct pe latura (BC).
Fie AM intersectat cu CD=N, DM intersectat cu AB in punctul P.
Demonstrati ca CD²=NC ·PB.

Answer 1

$\Delta{DCM}\sim\Delta{PBM}\Rightarrow \dfrac{DC}{PB}=\dfrac{CM}{BM}\ \ (1)$

$\Delta{NCM}\sim\Delta{ABM}\Rightarrow\dfrac{NC}{AB}=\dfrac{CM}{BM}\Rightarrow\dfrac{NC}{DC}=\dfrac{CM}{BM}\ \ (2)$

Din  (1)  si  (2) obtinem:

$\dfrac{DC}{PB}=\dfrac{NC}{DC}\Rightarrow DC^2=NC\cdot PB$