A necessary and sufficient condition for where and are points of the respective side lines of a triangle , to be concurrent is that
where all segments in the formula are directed segments.
Likewise, we find that
The trig version of Ceva's Theorem states that cevians are concurrent if and only if
This proof is incomplete. If you can finish it, please do so. Thanks!
We will use Ceva's Theorem in the form that was already proven to be true.
First, we show that if , holds true then which gives that the cevians are concurrent by Ceva's Theorem. The Law of Sines tells us that
Likewise, we get
- Suppose AB, AC, and BC have lengths 13, 14, and 15. If and . Find BD and DC.
If and , then , and . From this, we find and .
- See the proof of the concurrency of the altitudes of a triangle at the orthocenter.
- See the proof of the concurrency of the perpendicual bisectors of a triangle at the circumcenter.