How did inquisitions come to be used in dealing with cases of heresy?
The medieval inquisition started life in a series of papal bulls on heresy, in particular Lucius IIIs Ad abolendum of 1184 and Innocent IIIs Cum ex officii nostri of 1207. These defined the crime of heresy, imparted the duty of the church to root it out and equated it with treason against the state. It was always the case that the church was seeking to persuade rather than enforce but recognised that these efforts had failed and that stronger measures were required. Parallel to these developments the concept of inquisitio was being used widely in canon and civil cases as part of the professionalisation of law. Innocent IV formalised the procedures for all kinds of ecclesiastic enquiries as well as instituting the office of Inquisitor of Heretical Depravity.
