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
- 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.