The invention relates to a machining tool holder comprising alignment elements in a housing and clamping elements on the base of the housing, the elements being used to align and clamp a removable cutting tip comprising a lateral slide for the lateral deviation of a pressure force exerted by a screw belonging to the tool holder, the screw being guided such that it abuts obliquely against the lateral slide in order to push the slide, and consequently the tip, back towards a longitudinal alignment side which defines the housing and which serves as an alignment end stop for the associated alignment edge of the tip. In this way, the tip is oriented as desired in order to slide along the longitudinal side and abut against a rear alignment side.