Rangers won their quarter-final against Dundee 4-0 on Saturday at lunchtime before United claimed a dramatic 3-2 win over Ross County.
Celtic won the first of Sunday's quarter-finals, overcoming Greenock Morton 3-0.
Hibs and Caledonian Thistle then drew drew 1-1 at Easter Road.
Rangers v Celtic
Hibernian or Inverness Caledonian Thistle v Dundee United
Ties to be played on the weekend of 16/17 April.
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Rangers will play Celtic in the Scottish Cup semi-finals while Hibernian or Inverness Caledonian Thistle will play Dundee United.