Chet,
Your example does not answer the question. What if the first player checks and the next player mucks his hand with another player in the hand. That violates all rules of etiquette and does not offer protection for the other player, or players. A player should never muck his hand when the action is checked to him. They may check, or bet, never fold (unless the action is heads-up). In my opinion, check-muck is one of the most unethical, inconsiderate, improper actions that any poker player could ever make, when other players have yet to act.
Some casinos use the rule of last aggressive action on that betting round....and some casinos use the last act of aggression (even on a previous round). Example: Pre-flop Player A checks, and Player B checks. Player C bets 100, Players A and B both call.
Using TDA rule #11: After the river all players check. Player A is first to show hand.
Other cardroom ruling: Player C would show hand first, because the last bet made was initiated by Player C back in the first round of betting.
I much prefer the TDA rule, because it can be confusing at times to remember who bet way back in the earlier rounds.