In modern Euclidean geometry - the geometry specifically dealing with triangles and circles - a symmedian is a special line from one vertex to the opposite side of a triangle. A median is such a line - it connects one vertex to the midpoint of the opposite line. An angle bisector is another such line - it's the ray that bisects the angle inside the vertex, and it terminates in the other side.
The symmedian is the line that's the reflection of the median about the angle bisector. The median makes a certain (usually very small) angle with the angle bisector. The symmedian makes the same angle, only it's on the opposite side of the angle bisector.
You can see a picture of the three symmedians in the Wiki article referenced below.
Interestingly, the three symmedians of a triangle all meet in one point. That point is the symmedian point, which is labeled Kimberling center X(6). It has a lot of wonderful mathematical properties... which you can read about at your leisure.
Mister Chu: No, no, no....a symmedian is a creature from somewhere else who carries off our women in the manner of what happened (according to Athenaeus) to the original Syme who was the daughter of Ialysos and Dotis and got carried off by the sea god Glaucus on his way back from Asia. As one does. See?
- Wikipedia, "Symmedian,"