Screensharing crashes on login when using server name

I have three Macs, let's call them A, B, C, all running Snow Leopard. My main Mac is A. I have screen sharing set up on the other two Macs and until yesterday it worked fine. Now, I can still see B's screen from A as usual. But when I try to see C's screen using Go > Connect to Server and entering its name (C.local or vnc://C.local) or using Go > Home, clicking on C on the sidebar and choosing Share Screen, it does not work. The Screen Sharing app brings up a login dialog for C (which it has not done since I first connected to C long ago) and when I enter the correct login information, after a bit the Screen Sharing app crashes and leaves the login dialog hanging. I have to use Activity Monitor to quit NetAuthAgent to close the dialog.

I can connect to C from B with no problem, and I can connect to C from A if I use C's IP address (vnc://xxx.xxx.xxx.xxx) but not if I use its name. I can also see C from A using Chicken of the VNC. And I can connect to disks on C.

Of course, I have tried restarting A and C, and I have tried deleting the Screen Sharing prefs file (com.apple.ScreenSharing.plist) from my user Library on A (and restarting). I thought the problem might be with keychains, so I ran Keychain Access, and I noticed that there were vnc keychain entries for B but not for C. So I tried manually adding keychain entries for C. No differece. Then I deleted all keychain entries for C, which did not help.

Something on A that is specific to screen sharing C by name got clobbered, but I am at a loss what it might be or how to fix it.