logistic_guy
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Prove Ceva’s Theorem: If \(\displaystyle P\) is any point inside \(\displaystyle \Delta ABC\), then \(\displaystyle \frac{AY}{YC} \cdot \frac{CX}{XB} \cdot \frac{BZ}{ZA} = 1\).
Hint: Draw lines parallel to \(\displaystyle \overline{BY}\) through \(\displaystyle A\) and \(\displaystyle C\). Apply theorem \(\displaystyle 6.4\) to \(\displaystyle \Delta ACM\). Show that \(\displaystyle \Delta APN \sim \Delta MPC, \Delta CXM \sim \Delta BXP,\) and \(\displaystyle \Delta BZP \sim \Delta AZN\).
Hint: Draw lines parallel to \(\displaystyle \overline{BY}\) through \(\displaystyle A\) and \(\displaystyle C\). Apply theorem \(\displaystyle 6.4\) to \(\displaystyle \Delta ACM\). Show that \(\displaystyle \Delta APN \sim \Delta MPC, \Delta CXM \sim \Delta BXP,\) and \(\displaystyle \Delta BZP \sim \Delta AZN\).