Explicație pas cu pas:
AM este mediană: BM ≡ MC = ½•BC
QR || AM
ΔABM ~ ΔPBR
[tex]\frac{AB}{PB} = \frac{BM}{BR} \iff \frac{AB}{AB - PB} = \frac{BM}{BM - BR} \\ [/tex]
[tex]\frac{AB}{AP} = \frac{BM}{RM} \implies \frac{AB}{AP} = \frac{BC}{2 \cdot RM} \\ [/tex]
ΔQCR ~ ΔACM
[tex]\frac{QC}{AC} = \frac{CR}{CM} \iff \frac{QC - AC}{AC} = \frac{CR - CM}{CM} \\ [/tex]
[tex]\frac{QA}{AC} = \frac{RM}{CM} \implies \frac{QA}{AC} = \frac{2\cdot RM}{BC} \\ \iff \frac{AC}{AQ} = \frac{BC}{2\cdot RM}[/tex]
=>
[tex]\frac{AC}{AQ} = \frac{AB}{AP} \iff \boxed{\frac{AP}{AQ} = \frac{AB}{AC}} \\ [/tex]
q.e.d.
