Four-time major victor Rory McIlroy, whose morning 67 was highlighted by a 58-foot chip-in at 14, had three bogeys against two birdies over the first 10 holes in round three.
That left Woods five strokes off the pace of United States compatriot Koepka, who birdied five of his first nine holes to make the turn on 13-under par, three shots ahead of Americans Rickie Fowler and Gary Woodland with Australian Adam Scott, the 2013 Masters champion, and American Kevin Kisner another stroke adrift.
Then he drilled his tee shot at the 10th right down the middle.
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Ten players were within four shots of the lead, which includes defending champion Justin Thomas (68), Jason Day (67) and Stewart Cink, the 2009 British Open champion who played with Woods and matched his 66. The American shot a 66 to move to 12 under par for the tournament, two shots ahead of Adam Scott.
His nearest challenger is former Masters victor Adam Scott of Australia, who carded a fine five-under 65 for 10-under 200, two shots back.
Tiger Woods has a chance to win this thing at age 42, more than 10 years after his 14th and last major title and only three weeks after he held The Open lead with eight holes to play.
He is four strokes behind leader Brooks Koepka and in a tie for sixth. "Because of that, you got to make birdies". "If the guys that came this morning and played a certain amount of holes, they definitely need the recovery because it's going to be hard to recover fully from what we did today".
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Still, it is the first time since the 2005 Masters that Woods has shot consecutive rounds of 66 or better in a major championship. Finau is No. 13 in the standings and this is the last week to earn one of the eight automatic spots.
"I made sure on 17 I did". With a front pin on the seventh hole, he went some 25 feet beyond the flag and holed the putt, and then got up-and-down from a bunker on par-5 eighth.
"And tomorrow, not just myself, but everyone's going to have to shoot low rounds".
Woods was among 77 players who returned early on Saturday to complete the weather-delayed second round. He muscled it three feet past the hole, then pulled the comebacker off line and, after settling for a disappointing par, he briefly grabbed the putter with both hands, as if to bend it, before deciding better of it. He didn't register his first birdie until the 13th hole.
"They have been unbelievable", said Woods. That's exactly what happened on Friday at Bellerive Country Club.
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When I found out about this opportunity and what the manager wanted out of me I was happy. "I did not need any convincing to come here".
No one caught Woodland, meaning his 36-hole score of 130 stood as the PGA Championship record. He's been playing consistently good golf and, while he's yet to win, a victory seems to be on the way (whether that's today or at a major tournament in the future). After a par at No. 9, he rattled off three more birdies over his next four holes. "I didn't even think of it".