1. The most common form we see is
An easy-to remember form of this is rewriting the above as (a man and his dad hid a bomb in the sink!). This can be proved by applying the cosine rule to triangles ACD and then ABC:
2. If divides the side in the ratio ,
3. Similar to (2) but substituting ,
This and the previous form are conveniently proved using vectors. Writing the vector ,
Note that this is valid for any real , so may lie beyond segment .
4. Writing (3) as a quadratic in :
5. A symmetric form , where the following distances are taken as directed segments ( etc.)
Note that this is equivalent to which is (1).
Here are a few special cases of this formula applying form (3).
- (i.e. :
- is the midpoint of (): or (Apollonius’ theorem)
- is a third of the way along (closer to ) ():
- is the internal angle bisector of ():
- is the external angle bisector of (assume so ):