Given a triangle ABC with median AD. If AB = 1, AC = 2, and AD = BC, find the length of AD.
At least four different strategies might be considered. All may be worth trying.
Note: Some attempts to solve this problem have assumed that some auxillary lines produce similar triangles to facilitate the solution. IF you go this route, you need to PROVE your triangles are similar.
A GSP file if you want it.
* Adapted from Problem 255 in Http://www.8foxes.com
The 8foxes site was active for quite a while but it does not appear to have added new material since 2013.