A Clash of Active Kings

Hey Chess Friends,

I had a recent blitz game where both kings were in danger throughout. It reminded me of World Champion Steinitz's claim that the king was a strong piece, capable of defending itself. We reached a critical position early where my opponent, playing black, had to decide between two plausible moves. See the position below. 

Kd7 would be mated immediately by Qb7#.

That leaves the plausible options as Ke7 and Rxb8. They both reach very similar positions, so if you can't calculate all the way to a clear result, it's best to use the method of comparison. White will win a rook in either case, but the white queen will be stuck in a corner and the white king may be vulnerable to a perpetual check. At the very least it can be chased all the way to h3 or h4.. Here are the two possible variations.

Rxb8 line:

19... Rxb8 20.Qxb8+ Ke7 21. Qxh8 Qxc2+ 22. Kf3 Qe4+ 23. Kg3 Qg6+ 24. Kh4 Black to move in the position below.

Ke7 line:

19... Ke7 20. Rxa8 Rxa8 21. Qb7+ Kf6 22. Qxa8 Qxc2+ 23. Kf3 Qe4+ 24. Kg3 Qg6+ 25. Kh4 Black to move in the position below.

In both variations white could have retreated the king back to the f-file and achieve a draw, but considering the material situation, that would be a success for black. 

So, the real question for black is where is the king better, f6 or e7. It's hard to calculate far enough to immediately tell which line is better, so simply comparing which square is better for the king is a good way to make your choice. It turns out that one of positions above is winning for white and the other winning for black. Before you scroll down, try to determine which one black can win and how. There aren't good checks in either position so black needs to find a creative idea.

.....

Did you try to figure it out?

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It turns out that with the king on e7 black is just lost. There's no way to continue the attack and white will convert the extra rook into a win. In the Ke7 line black has an unexpected winning move. Kf5! which forces mate. The active king is perfectly safe and threatens Qg4# as well as Be7+. White is helpless. The position with both kings on the 4th and 5th rank deserves another look.

In case you're interested, here's the whole game. My opponent didn't find the correct defense.