C172 INSTRUCTOR AND STUDENT HAVE AN NMAC ON FINAL APCH.

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I WAS INSTRUCTING A PVT PLT STUDENT; REMAINING IN THE CONVENTIONAL L-HAND TFC PATTERN FOR RWY XX. TFC HAD VARIED BTWN RWY YY AND RWY XX THROUGHOUT THE DAY WITH WINDS SOMEWHAT IN BTWN. THE STUDENT WAS MAKING MOST OF THE RADIO CALLS BUT I WAS HELPING OUT AT THE TIME BECAUSE ANOTHER AIRPLANE WAS IN THE PATTERN TO LAND ON RWY YY. WE CLRLY COMMUNICATED WITH THAT PLT AND SEEMED TO HAVE NO PROBS GETTING GOOD RADIO RECEPTION FROM HIM; WE VISUALLY ASCERTAINED EACH OTHERS' WHEREABOUTS; AND WERE NO FACTOR WITH HIM. UPON TURNING BASE FOR THE NEXT LNDG; I NOTICED THE HELI OPERATING TO THE R SIDE OF RWY XX. I HAD HEARD NO RADIO CALLS FROM HIM; AND BECAUSE I KNOW THE FLT SCHOOL THAT OPERATES THIS HELI AND KNOW THAT THEY ARE TYPICALLY VERY GOOD AT COMMUNICATING INTENTIONS WHEN GETTING CLOSE TO FIXED WING TFC; I ASSUMED THAT THEY MUST JUST BE MANEUVERING TO THE SIDE OF THE ARPT BUT WERE NOT INBOUND FOR LNDG. THIS IS A COMMON PRACTICE FOR THEM; AND I HAVE GOTTEN USED TO SEEING THEM OFF TO THE SIDE OF THE RWY. I MADE SURE MY STUDENT MADE A RADIO CALL 'TURNING BASE TO FINAL' JUST TO ENSURE THEY KNEW WE WERE THERE. AFTER A FEW MORE SECONDS; THE HELI SEEMED TO BE CONVERGING WITH US APCHING A FINAL PATH FOR RWY XX. JUST TO BE SAFE; I THEN MADE A RADIO CALL ANNOUNCING SHORT FINAL FOR RWY XX. SINCE THERE WAS NO RESPONSE FROM THE HELI; I ASSUMED THEY KNEW THEY WERE NOT A FACTOR AND I CONTINUED. I SHOULD PROBABLY HAVE MANEUVERED CLR AT THAT POINT SINCE I HAD NO CONFIRMATION OF THEIR INTENTIONS OR THEIR KNOWLEDGE OF MY WHEREABOUTS; BUT WAS STUCK IN MY ASSUMPTION THAT THEY MUST BE PLANNING TO REMAIN CLR. AFTER A FEW SECONDS IT WAS CLR THAT WE WERE OVERTAKING THEM UNDERNEATH; BUT AT THAT POINT A TURN SEEMED LIKE A WORSE CHOICE THAN SIMPLY ACCELERATING OUR DSCNT TO PUT MORE VERT DISTANCE BTWN US. WE FLEW UNDERNEATH THE DSNDING HELI AND LANDED.