INSTRUCTOR WITH A STUDENT ON BASE LEG TAKES EVASIVE ACTION CLB TURN TO AVOID TFC NOT OBSERVED BUT SOUNDING CLOSE BY TWR CTLR RESULTS IN NMAC.

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FLT INSTRUCTOR WITH A JAPANESE STUDENT (1.0 HRS SOLO). I WAS TO EVALUATE MY STUDENT'S ABILITY TO ENTER AND EXIT HIGH DENSITY TFC PATTERNS (NON-RADAR). MY ACFT WAS R DOWNWIND ABEAM THE APCH END OF RWY 21. ANOTHER ACFT RPTED A 2 MILE R BASE. THE TWR CTLR INSTRUCTED US TO FOLLOW THAT TFC WHICH REQUIRED US TO MAKE A 45 DEG CORRECTION TO THE L. THE STUDENT RESPONDED THAT HE WAS LOOKING FOR THE TFC. I SPOTTED THE TFC AND INSTRUCTED MY STUDENT TO LOOK FURTHER TO HIS L FOR THE TFC. I THEN HEARD ANOTHER ACFT RPT 7.2 DME ON THE LOC (RWY 21). THE CTLR THEN TOLD US TO WIDEN OUT EVEN MORE TO FOLLOW THE TFC ON BASE (NOT YET IN SIGHT TO MY STUDENT) AND TOLD THE OTHER ACFT TO RPT 4.2 DME. MY STUDENT RPTED TFC IN SIGHT (ON BASE). WE WERE CLRED TO LAND BEHIND THAT TFC. WE WERE ON A WIDE R BASE ABOUT A MI FROM THE LOC WHEN THE CTLR ASKED THE POS OF THE ACFT ON FINAL. THEY RESPONDED 4.4 DME. WE WERE THEN INSTRUCTED TO FOLLOW THIS TFC. I INFORMED THE TWR THAT THE TFC WAS NOT IN SIGHT. I COULD TELL BY HIS VOICE THAT HE WAS UNCOMFORTABLE WITH WHAT WAS HAPPENING WHEN HE ISSUED A SECOND TA ABOUT THE TFC ABOUT 4.3 DME. I AGAIN REPLIED 'NEGATIVE CONTACT.' HE IMMEDIATELY ISSUED INSTRUCTIONS TO 'TURN L NOW!!' I TOOK THE CTLS AND MADE A 60 DEG BANK CLBING TURN AT APCH SPD. AFTER I ROLLED OUT OF THE TURN I SAW THE ACFT ON THE LOC DIRECTLY AHEAD BUT SLIGHTLY LOWER. WE PASSED WITHIN 100 FT OF EACH OTHER. HAD WE BEEN AT THE SAME ALT THERE WOULD HAVE BEEN NO TIME TO AVOID A HEAD ON MIDAIR COLLISION. THE REMAINDER OF THE FLT WAS UNEVENTFUL. THE NEXT DAY I CONTACTED THE CTLR I SPOKE WITH ON THE RADIO. WE BOTH UNDERSTAND EACH OTHER'S POS AND EACH EXPLAINED OUR SIDE OF THE STORY; AND WE RESOLVED THE SITUATION.