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Donald J. Trump was declared the winner of the Nevada caucuses on Tuesday night, according to The Associated Press, gaining a third consecutive victory in an early-voting state and strengthening his position in the Republican presidential race before the wave of Super Tuesday elections on March 1.

Turnout in Nevada was reported high compared with previous caucuses. Mr. Trump was seen as a favorite going into the contest, but his victory still serves as a setback for his chief competitors, Senators Ted Cruz of Texas and Marco Rubio of Florida, who must now try to break Mr. Trump’s winning streak in the larger states that vote in the coming weeks.

For Mr. Trump, the outcome in Nevada is another sign of his campaign’s durability and the breadth of his appeal: He has now handily won primary elections in New England and in the South, and a caucus fight in the far West. He won over independent voters in New Hampshire and evangelicals in South Carolina, and prevailed in Nevada, where Mormon voters and rural activists wield influence.

This latest triumph may only encourage Mr. Trump in the brash campaign style that has alienated many Republican officials and mainstream voters. In the two days leading up to the Nevada caucuses, Mr. Trump called Mr. Cruz a liar and threatened to deliver vicious attacks on Mr. Rubio as well.

At a rally in Las Vegas on Monday evening, Mr. Trump ridiculed a protester in his audience and told supporters that he would have liked to “punch him in the face.”

His supporters in Nevada were jubilant on Tuesday night. Holding Trump signs and flags and a few Bud Lights, the crowd at the Treasure Island casino erupted into a minute-long cheer when Mr. Trump was projected the winner on a CNN broadcast, and chanted the candidate’s name.

But the chants quickly turned to boos as soon as the network mentioned Mr. Cruz.

The results are likely to reinforce the sense among national Republican leaders that only direct confrontation can block Mr. Trump from claiming the party’s nomination, because none of the party’s most powerful voting blocs seems likely to thwart him on its own.