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Brier, the 2007 Volvo China Open winner, overtook overnight leader Retief Goosen with a bogey-free round on a much calmer day after strong winds on Thursday had caused havoc on the desert course.
Goosen and Brier were neck and neck over the first nine holes in the second round, but the South African slipped down later with a bogey and a double bogey that saw him shoot a 71 for an aggregate of four-under-par 140, three shots behind the leader.
Fellow South African Darren Fichardt was in sole second spot after shooting a 68 for an aggregate of 138, while Denmark's Thomas Bjorn, Sweden's Robert Karlsson and England's Richard Finch were bunched a shot behind on 139.
But it was Kaymer's fight to stay in the hunt to become only the second German world number one after Bernhard Langer that took centre stage on a day when normal service resumed following a chaotic first day, when only eight players managed to better par.
With current world number one Lee Westwood missing the halfway cut after aggregating a four-over-par 148, Kaymer can take over at the top of the rankings even if he finishes second.
Kaymer, who shot a birdie-free 77 in the first round, retrieved his position with a bogey-free effort of 72 in the second, giving him an aggregate score of 147.
But the 26-year-old, who won in Abu Dhabi for the third time last month, admitted that he was not playing his best at Doha Golf Club.
"To be honest, I really struggled here," said Kaymer.
"I don't know why. I play good golf but I can't score."
"The thing is that the tournament here is very tight and no-one is running away, and I am only seven shots off the pace.
"And I heard the weather is going to be good tomorrow (Saturday), so I will play more aggressive in order to make a few more birdies.
"We'll see if I have a chance on Sunday."
Scoreboard:
137 - Markus Brier (AUT) 71-66
138 - Darren Fichardt (RSA) 70-68
139 - Thomas Bjorn (DEN) 74-65, Robert Karlsson (SWE) 70-69, Richard Finch (ENG) 70-69
140- Thomas Aiken (RSA) 71-69, Paul Lawrie (SCO) 74-66, Retief Goosen (RSA) 69-71
141 - Robert Rock (ENG) 73-68
142 - Victor Dubuisson (FRA) 74-68, David Drysdale (SCO) 71-71, Louis Oosthuizen (RSA) 73-69, David Lynn (ENG) 73-69, Steven O'Hara (SCO) 73-69
143 - Seve Benson (ENG) 71-72, Thongchai Jaidee (THA) 74-69, Felipe Aguilar (CHI) 77-66, Oliver Fisher (ENG) 71-72, Oliver Wilson (ENG) 74-69, Bradley Dredge (WAL) 72-71, Ian Poulter (ENG) 72-71
144- David Howell (ENG) 75-69, Jeev Milkha Singh (IND) 73-71, Kenneth Ferrie (ENG) 75-69, Alvaro Quiros (ESP) 75-69, Ignacio Garrido (ESP) 73-71, Niclas Fasth (SWE) 70-74, Simon Dyson (ENG) 72-72, Jeppe Huldahl (DEN) 74-70, Jean-Baptiste Gonnet (FRA) 73-71, Mark Brown (NZL) 75-69, Mark Foster (ENG) 73-71
