Endgame study ALWAYS pays off.
A Rook Endgame Swindle

The immediate Rh4 would lose the game as you pointed out magicmaster. Rg4+ followed by either Rf4+ or Rh4+ would keep the draw.
The immediate Rh4 would lose the game as you pointed out magicmaster. Rg4+ followed by either Rf4+ or Rh4+ would keep the draw.
From a recently completed game. Black has just played 50...Re2, demonstrating that my intended plan of marking time with the rook on the 4th rank until swapping pawns into a Philidor position isn't going to work. The threat is Re4, winning the h-pawn or forcing an exchange into a losing king-and-pawn endgame. What would you play?