Help with proof involving Cevians.

**jpatrie** · February 7th 2011, 11:15 AM

Given $\Delta ABC$ as shown with $AF=\frac{1}{4} AB$ , $AE=\frac{1}{4} AC$ and $\overline{BE} \cap \overline{CF}=0$ prove $AO$ hits $\overline{BC}$ at its midpoint.

**Ithaka** · February 7th 2011, 11:33 AM

Originally Posted by jpatrie

Given $\Delta ABC$ as shown with $AF=\frac{1}{4} AB$ , $AE=\frac{1}{4} AC$ and $\overline{BE} \cap \overline{CF}=0$ prove $AO$ hits $\overline{BC}$ at its midpoint.

Let {D}= AO intersection BC

Use Ceva Theorem:

$\frac{AF}{FB} * \frac{BD}{DC} * \frac{CE}{EA}=1$

Simplify and obtain the ratio BD : DC

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