NMAC BTWN 2 C172'S APCHING THE TFC PATTERN AT 3 MI FROM A CTLED ARPT. TA ISSUED.

Narrative

UPON RETURNING TO SGR; MY STUDENT MADE INITIAL CALL TO SGR TWR THAT WE WERE 8 MI S INBOUND (DISTANCE DETERMINED BY CHART LOCATION). TWR INSTRUCTED US TO RPT 4 MI SE FOR THE L DOWNWIND TO RWY 17. SHORTLY THEREAFTER; WHILE WE WERE APPROX 7 MI OUT; ACFT #1 RPTED 3 MI SE. I BEGAN LOOKING FOR ACFT #1 ABOUT 3 MI AHEAD OF OUR POS (I DID NOT HAVE HIM IN SIGHT). WE PASSED 4 MI (DISTANCE DETERMINED BY CHART POS) AND MY STUDENT RPTED TO SGR TWR SHORTLY THEREAFTER AT 3 MI (LATE CALL DUE TO RADIO TFC). I STILL DID NOT HAVE ACFT #1 IN SIGHT; BUT EXPECTED HIM TO BE 3 MI AHEAD BASED ON HIS PREVIOUS POS RPT. AT THIS TIME I ALERTED MY STUDENT OF HIS ALT; WHICH WAS 900 FT MSL (PATTERN ALT IS 1000 FT MSL). TWR THEN CALLED ME SAYING THAT TFC WAS TO MY R. A FEW SECONDS LATER; I NOTICED ACFT #1 MERGING AND OVERTAKING US FROM R TO L (HDG DIFFERENCE WAS ABOUT 20 DEGS). VERT DISTANCE WAS 40 FT AS HE PASSED DIRECTLY UNDERNEATH US. I TOOK CTL OF THE AIRPLANE; AND INITIATED A CLBING R TURN AWAY FROM THE TFC; FALLING INTO A WIDE DOWNWIND OUTSIDE AND BEHIND ACFT #1.