WASHINGTON — The House managers prosecuting President Trump for high crimes and misdemeanors have failed so far to persuade Senate Republicans to let them call new witnesses in his impeachment trial. But in their own way, they have come up with a star witness they can bring to the floor: Mr. Trump himself.

Barred at this point from presenting live testimony, the managers have offered up the president as the most damning witness against himself, turning his own words against him by quoting from his public remarks, citing accounts of private discussions and showing video clips of him making audacious statements that the House team argues validate its case.

Thanks to screens set up in front of the senators, Mr. Trump’s voice has repeatedly echoed through the Senate chamber the past three days. There he was on the South Lawn of the White House publicly calling on Ukraine to investigate a campaign rival, former Vice President Joseph R. Biden Jr. There he was calling on China to go after Mr. Biden, too. There he was declaring that he would willingly take foreign help to win an election. And there he was back in 2016 calling on Russia, “if you’re listening,” to hack into Hillary Clinton’s email.

The strategy seeks to capitalize on Mr. Trump’s astonishingly unfiltered approach to politics, which has led him again and again to say openly what other presidents with more of an understanding of the traditional red lines of Washington — or at least more of an instinct for political self-preservation — would never say in front of a camera.