WASHINGTON (NYTIMES) - President Donald Trump and Fox News have a complicated relationship. Election Day did not help.

The cable news channel that kick-started Mr Trump's political career was suddenly in the position of signalling its potential end.

The network's early call of Arizona on Tuesday night (Nov 3) for Mr Joe Biden infuriated Mr Trump and his aides, who reached out publicly and behind the scenes to Fox News executives about the call.

The network held firm - even as two of its biggest stars, Ms Laura Ingraham and Ms Jeanine Pirro, attended Mr Trump's defiant early morning speech in the East Room of the White House.

The election-night split screen underscored the fine line that Fox News anchors and opinion hosts have been walking.

By Wednesday afternoon, Fox News was the closest of any major network to calling the presidential race for Mr Biden - not the outcome that many fans of its pro-Trump programming may have wanted.

Fox News was also the only major cable network to carry a news conference on Wednesday held by the president's lawyer, Mr Rudy Giuliani, who was making baseless claims of election fraud.

But the channel promptly cut away to announce a major development: It projected a win in Michigan for Mr Biden, placing him at the doorstep of the presidency, according to Fox's projections.

And shortly after Mr Bret Baier, the network's chief political anchor, emphasised to viewers on Wednesday that Mr Trump's threatened litigation could throw the race into doubt - even if Mr Biden was projected to win 270 electoral votes - Fox News' politics editor, Mr Chris Stirewalt, threw cold water on some of the Trump campaign's baseless claims.

"Lawsuits, schmawsuits," Mr Stirewalt said. "We haven't seen any evidence yet that there's anything wrong."

Fox News has long occupied an unusual position in the Trump orbit. The network is home to some of the President's most vociferous defenders, including Mr Sean Hannity, Ms Ingraham, and the hosts of Fox & Friends.

But Mr Trump frequently takes potshots at its news division and polling operation.

"Fox has changed a lot," Mr Trump said on Tuesday morning on Fox & Friends.

"Somebody said, 'What's the biggest difference between this and four years ago?' I say, 'Fox.'"

The President is a vociferous viewer and constant critic, praising preferred hosts by first name at rallies ("Jeanine!" "Tucker!") and dialling up the network's chief executive, Ms Suzanne Scott, to complain about coverage.

He has hired (and fired) former network personnel; belittled its hosts while also agreeing to interviews; and relied on Mr Hannity's political advice while bashing news anchors like Mr Chris Wallace and Mr Shepard Smith, who left the network for CNBC.

In the wake of Tuesday's Arizona call, a mixed view of Fox News had spread to some of Mr Trump's allies.

Governor Ron DeSantis of Florida, a Republican who rose to fame on the strength of Fox News guest appearances, bashed the network for what he deemed an insufficiently swift projection of a Trump win in his home state.

"For Fox to be so resistant to calling Florida and yet jumping the gun on Arizona, I just thought was inexplicable," Mr DeSantis told reporters in Tallahassee on Wednesday.

"I don't think that that was done without some type of motive, whether it's ratings, whether it's something else."

In fact, members of Fox News' decision desk said repeatedly that the network's polling team - which reports to the news division and is sequestered on election night - was merely adhering to a rigorous analysis.

The network's data team, led by Mr Arnon Mishkin, relies on a proprietary model that draws on data from The Associated Press.

Still, some Fox News personalities speculated whether Arizona would remain in Mr Biden's column.

"There may be some tightening there," Mr Baier said on Wednesday, summarising arguments from the Trump campaign, while Mr Bill Hemmer used an interactive map to conjure ways Mr Trump could eke out a win in Pennsylvania.

But when Mr Hemmer asked if the network might consider reversing the Arizona call, Mr Stirewalt laughed. "Not that I see," he said.

Mr Wallace also offered a grim prognosis for the President.

"It's real simple math now," he said, shortly after Fox News projected that Mr Biden would win Wisconsin.

Pointing to Mr Biden's advantages in Nevada and Michigan, he said: "If he just holds on to his lead in those two states, he's the 46th president of the United States." (Fox News would call Michigan just over an hour later.)

Mr Hannity did not appear on Fox News on election night, but he returned on Wednesday evening, echoing some of the President's talking points about the integrity of the vote count.

He stopped short, though, of Mr Trump's baseless claim of outright "fraud".

"Do you trust what happened in this election?" Mr Hannity asked viewers. "Do you believe these election results are accurate? Do you believe this was a free and fair election? I have a lot of questions."

Mr Hannity had few specific arguments, tossing in a reference to "dead people", and at times his monologue sounded like a regular episode of his programme, not a post-election special.

His lead-in, Mr Tucker Carlson, also spoke ominously about the vote results while avoiding an outright embrace of Mr Trump's baseless claims about winning states that had yet to be called.

"Many Americans will never again accept the results of a presidential election," Mr Carlson said at one point.

The network set a record for the highest Election Day prime-time viewership totals in cable news history, according to Nielsen.

Fox News drew 14.1 million viewers between 8pm and 11pm.

Its next-closest competitor, CNN, drew nine million in the three-hour block.

Each of the three top broadcast networks carried less than half of Fox's audience, with ABC, at 6.3 million viewers, leading that group.

This followed a delirious ratings run for Fox News in the weeks leading up to the election.

In October, Fox News averaged 4.9 million viewers in prime time, up 85 per cent from a year earlier and far higher than MSNBC, which finished second with 2.7 million viewers.

Tucker Carlson Tonight earned an average of 5.4 million viewers in October, the highest monthly average for any show in the history of cable news.

And Fox News even scored higher viewership totals than any of the broadcast networks during the two presidential debates and the vice-presidential debate.

Mr Lachlan Murdoch, Mr Rupert Murdoch's elder son and the executive chairman of Fox News' parent company, was asked on a Tuesday earnings call if a prospective Biden victory might rein in the channel's ratings success.

He pointed out that Fox News had dominated cable news rivals through "different administrations and different political cycles".

Mr Murdoch added: "We fully expect to be No. 1."

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