a)
[tex]\it \left.\begin{aligned}\it ABCD-\ paralelogram\ \Rightarrow AB||CD\\ \\ E\in AB\end{aligned}\right\} \Rightarrow AE||CD\ \ \ \ \ (1) \\ \\ \\ \left.\begin{aligned}\it ABCD-\ paralelogram\ \Rightarrow AB=CD \Rightarrow \dfrac{AB}{2}=\dfrac{CD}{2}\\ \\ \it AE=\dfrac{AB}{2}\end{aligned}\right\} \Rightarrow AE=\dfrac{CD}{2}\ \ \ \ \ (2) \\ \\ (1),\ (2) \Rightarrow AE-\ linie\ mijlocie\ \^{i}n\ \Delta FDC \ \ \ \ \ \ (3)[/tex]
b)
[tex]\it (3) \Rightarrow FE=EC\\ \\ Dar,\ AE=EB[/tex]
Prn urmare, diagonalele patrulaterului AFBC se înjumătățesc, deci
AFBC - paralelogram ⇒ AC || FB
[tex]\it c)\ AE=EB,\ AO=OC \Rightarrow OE-\ linie \ mijlocie\ \^{i}n\ \Delta ABC \Rightarrow[/tex]
[tex]\it \Rightarrow OE||BC||AF,\ iar\ G\in OE \Rightarrow EG||AF\\ \\ Dar,\ AE=EB \Rightarrow EG-\ linie\ mijlocie\ \^{i}n\ \Delta BAF \Rightarrow AF=2EG[/tex]
