Let $ABC$ be a triangle, and $a$ lie on $BC$, etc.. Given that $a$ cuts $BC$ in the proportion $k:(1-k)$, and $b$ cuts $CA$ in $l:(1-l)$, if $Cc$ passes through the intersection $Aa\cap Bb$, and cuts $AB$ in $m:(1-m),$ what is the relationship between $k,l,$ and $m$? 
Geometric reasoning would be appreciated over coordinates; if coordinates make it easier, trilinear coordinates are appreciated above the traditional orthogonal. But hey, beggars can’t be choosers!
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2 
- 2$\begingroup$ Are you familiar with Ceva's Theorem? $\endgroup$Blue– Blue2025-03-24 15:09:38 +00:00Commented Mar 24 at 15:09
- 1$\begingroup$ @Blue Thanks for the tip, I'm not much of a geometer as it turns out lol! $\endgroup$Alexander Conrad– Alexander Conrad2025-03-24 15:11:56 +00:00Commented Mar 24 at 15:11
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1 Answer
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Ceva's theorem would have that $$\boxed{\boxed{klm=(1-k)(1-l)(1-m).}}$$ (Credit to @Blue.)