I have three landline phones and one phone number. One is a plug-in answering machine connected to jack #1 (closest to the incoming line from the phone company). All three rang until recently when the phone in jack #3 did not ring; the others ring fine. I thought the phone in jack #3 was bad but it works fine in jack #2 and the phone that was in jack #2 does not ring in jack #3. I even switched phone cords, and then switched DSL filters with the same result: jack #3 suddenly stopped supporting a ring. I can still use the phone in jack #3 to dial out or to pick up incoming calls if I hear the ring from the other phones, but I would like the phone connected to jack #3 to work. Any ideas how to solve this?

Look inside where you plug your phone into for green or black discoloration, and check the line cord.
I would bet Jack.
If either are bad, replace cord and jack.

All contact points are bright clean copper: the cords, the insets in the phone, and the phone jack itself. There had been a high-tech gel in the jack that the tech has used to prevent shorts, which had spilled over the contacts. I disassembled the jack, cleaned off the contacts, resassembled it with no improvement

If that jack internal and external connections are good, you may have a problem with a connection at one of the other jacks. It may go directly to the outside box, which should be checked as well. You may see 1 pair with a bad connections.