A C172 TRAINING FLT HAS AN NMAC WITH ANOTHER GA ACFT IN THE TFC PATTERN AT APA; CO.

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MY STUDENT AND I HAD RETURNED TO THE APA ARPT AT THE END OF A TRAINING FLT AND AT THE TIME OF THE INCIDENT HAD PERFORMED 2 TOUCH-AND-GOES ON APA RWY 17L. AFTER THE SECOND TKOF; WE INFORMED THE TWR CTLR THAT THE NEXT LNDG WOULD BE A FULL STOP. ON L DOWNWIND APPROX ABEAM THE TWR (1; IN DIAGRAM); WE WERE INSTRUCTED TO CONTINUE OUR DOWNWIND TO ACCOMMODATE OTHER LNDG TFC I BELIEVE STATED TO BE OVER CHERRY CREEK RESERVOIR AND THAT TWR WOULD CALL OUR BASE TURN. STILL ON DOWNWIND AND APPROX OVER ARAPAHO ROAD (2); ABOUT 1 MI N OF THE RWY 17L THRESHOLD 'LATITUDE' I SAW TFC LOWER AT MY 9-10 O'CLOCK POS ON FINAL (3) AND I RPTED THIS TFC USING ALL OF THAT LOCATION INFO. I DO NOT REMEMBER THE TWR'S RESPONSE; BUT I REMEMBER BEING CONFIDENT THAT THAT WAS OUR TFC TO FOLLOW AND AFTER ALLOWING MAYBE 5 SECONDS TO PASS; I INSTRUCTED MY STUDENT TO TURN BASE (4) AND BEGIN OUR DSCNT. A FEW SECONDS LATER THE TWR QUESTIONED WHAT I WAS DOING AND INSTRUCTED SOMETHING LIKE 'KEEP THE L TURN COMING.' I INTERPED THIS TO MEAN THAT I WAS TO CONTINUE THE L TURN BASE TO FINAL AND PROCEED DOWN FINAL. SHORTLY THEREAFTER; MY STUDENT AND I SAW ANOTHER ACFT ABOVE AND TO THE W ON A R BASE TURNING FINAL ABOVE US (5). SOMEHOW THIS WAS APPARENTLY THE TFC WE WERE TO HAVE FOLLOWED. UPON A QUICK QUERY OF THE TWR AS TO WHAT THEY WANTED ME TO DO TO RESOLVE THE CONFLICT; THEY TOLD ME TO 'KEEP THE L TURN COMING' OR SOMETHING TO THAT EFFECT. I ASKED IF THEY WANTED ME TO CONTINUE THE TURN TO THE E AND THEY SAID 'YES.' IN FACT; AFTER A FURTHER INTERCHANGE; I COMPLETED A L 360 DEG TURN AND WAS CLRED TO LAND ON RWY 17L. THERE WAS DEFINITELY SOME MISCOM HERE. FIRST OF ALL; I DID NOT FOLLOW TWR'S COMMAND THAT THEY WOULD CALL MY BASE TURN; BUT AFTER RPTING THE TFC ON FINAL AS I DID; I WAS UNDER THE IMPRESSION THAT THAT WAS OUR TFC TO FOLLOW. I AM NOT SURE WHY TWR DIDN'T SAY SOMETHING ABOUT CONTINUING MY DOWNWIND FURTHER TO THE N AFTER MY RPT. THE COMMAND TO 'KEEP THE L TURN COMING' WAS NOT VERY CLR. A COMMAND OF 'DO A L 360 DEG TURN' OR 'CONTINUE THE L TURN TO AN EASTERLY HDG' WOULD HAVE BEEN MUCH CLRER. WHAT WILL I DO DIFFERENTLY IN THE FUTURE? I AM IN THE HABIT OF AND I TEACH IDENTING TFC POINTOUTS BY RPTING TFC DIRECTION FROM ME ('TFC 10 O'CLOCK...'); RELATIVE ALT ('...LOWER...') AND DIRECTION IF APPROPRIATE ('...ON FINAL.'); NOT JUST 'TFC IN SIGHT.' IN THE FUTURE I WILL PREFACE THIS WITH 'TFC TO FOLLOW IN SIGHT; 10 O'CLOCK POS; LOWER; ETC.' THIS WAY; ATC WILL KNOW THAT IT IS MY UNDERSTANDING THAT I AM TO FOLLOW THAT ACFT. WHAT SHOULD APA TWR HAVE DONE DIFFERENTLY? THE USE OF MORE STANDARD PHRASEOLOGY WOULD HAVE HELPED. AND I STILL DON'T KNOW IF THE TFC THAT I INTERACTED WITH WAS IDENTED EARLIER BY TWR AS BEING MY TFC TO FOLLOW OR IF THEY MISCALLED MY TFC TO FOLLOW EARLIER. ONLY A PLAYBACK OF THE TAPES WOULD SOLVE THAT MYSTERY.