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Both were enduring characters in New York, their names in headlines, their faces on television. Both began their lives outside Manhattan — one was born in Brownsville, Brooklyn, the other in Jamaica, Queens — and both worked their way toward its irresistibly sizzling spotlight. Both were flamboyant attention-getters who found fame on television, Donald J. Trump as a reality-show host, the Rev. Al Sharpton as a cable news commentator.

[Related: Trump lashes out at Al Sharpton, saying he “hates whites.”]

From time to time, they crossed paths, drawing energy from each other even as foes, as Mr. Trump made claims that Mr. Sharpton challenged — about five black and Latino teenagers who were charged with raping a white jogger in Central Park in the 1980s, and about whether Barack Obama was born in the United States.

But sometimes they were friends in the way that public figures in New York can be. Mr. Trump cut the ribbon at Mr. Sharpton’s National Action Network annual convention in 2002, returning four years later to pose with Mr. Sharpton, the Rev. Jesse Jackson and the singer James Brown.

“Different tune now,” Mr. Sharpton observed on Monday.

The two have once again found themselves convenient foils, after President Trump on Saturday denounced Representative Elijah E. Cummings, calling the African-American congressman, a Democrat who represents much of Baltimore, “racist,” and his district a “disgusting, rat- and rodent-infested mess.”