An involutory automorphism of an inversive plane with exactly two fixed points is called an harmonic involution. In this paper we study inversive planes with su‰ciently many harmonic involutions satisfying an analog of the theorem about the three reflections.2000 Mathematics Subject Classification. 51B10 Die von H P :¼ fs A H j P s ¼ Pg erzeugte Gruppe hH P i operiert transitiv auf PnfPg, daher operiert hHi zweifach transitiv auf P.