When a .ctx route is saved the .ctx extension isn't included.
You have to manually add it (in Windows, Tools > Folder Options... > View > UNCHECK, "Hide extensions for known file types".)
After adding .ctx to the end of the file name, when you ReCheck "Hide extensions for known file types" the extension is still visible.
All the above isn't the case with .GPX files
Comments
In theory it saves it as cycle_travel.tcx - but the code that produces it is a bit browser-dependent. What browser and version are you using?