Ceva三角形中的乘积、差与商

dlsh

\[\frac{\overline{C D}}{\overline{D B}}=\frac{(\overline{C F} \cdot \overline{P C})-(\overline{C E} \cdot \overline{A C})}{(\overline{B E} \cdot \overline{B P})-(\overline{B F} \cdot \overline{B A})}\]

Ly-lie

如图，由梅氏定理知：\[ \frac{PE}{EB}\cdot \frac{BC}{CD}\cdot \frac{DA}{AP}=1 \]\[ \frac{PF}{FC}\cdot \frac{CB}{BD}\cdot \frac{DA}{AP}=1 \]两式相除得$ \frac{BD}{CD}=\frac{FP\cdot BE}{EP\cdot CF} $，故\[ \frac{BP\cdot BE-BF\cdot BA}{CP\cdot CF-CE\cdot CA}=\frac{BE(BP-BU)}{CF(CP-CV)}=\frac{UP\cdot BE}{VP\cdot CF}=\frac{FP\cdot BE}{EP\cdot CF}=\frac{BD}{CD} \]