ZFW CTLR DESCRIBED CONFLICT BETWEEN IFR AND UNKNOWN VFR TFC; CITING A NUMBER OF POTENTIAL CONFLICTIONS; ENDING WITH A TFC PATTERN EVENT.

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THIS EVENT OCCURRED ON TXK-LO AT ZFW. I WAS PROVIDING OJT TO ANOTHER CPC (TRAINING NOT A FACTOR). WE WERE WORKING ACFT Y (B1900) DFW..ELD AT 10000 FT. THE FLT WAS EBOUND. WE OBSERVED A FAST MOVING VFR INTRUDER TARGET INDICATING 10000 FT; AT 10 O'CLOCK POS TO THE ACFT Y. WE CALLED THE TFC; WHICH PASSED ABOUT 1 - 1 1/2 MI L TO R SLIGHTLY CONVERGING IN FRONT OF THE ACFT Y. THE PLT OF ACFT Y RPTED THE TFC IN SIGHT. WHEN THE VFR WAS ABOUT 5 O'CLOCK POS; 6 MI BEHIND ACFT Y; IT BEGAN A WIDE L TURN -- ALT STILL INDICATING 10000 FT (OR WITHIN A FEW HUNDRED FT). AFTER THE TURN; WE AGAIN ISSUED TFC TO THE ACFT Y. 'LOOKING' WAS THE REPLY. THE VFR CONTINUED NEBOUND ON A CONVERGING COURSE WITH ACFT Y. THE CPC-IT AGAIN CALLED TFC AND ISSUED A DSCNT TO ACFT Y. THE PLT RPTED THAT TFC IN SIGHT. I ASKED WHAT TYPE ACFT THE OTHER TFC WAS; AND THE PLT OF ACFT Y RPTED IT TO BE A LANCE 4. THEN; THE VFR TFC MOVED SO CLOSE TO THE ACFT Y -- VERT AND HORIZ -- THAT WE ACTUALLY LOST RADAR RETURNS ON BOTH ACFT. THE PLT'S VOICE WAS SHAKY; SO I OVERRODE THE CPC-IT AND AUTH THE ACFT Y TO MANEUVER L OR R AND AUTH A DSCNT TO 2000 FT. THE VFR TFC THEN MOVED AWAY NEBOUND FROM ACFT Y. I RECALLED WORKING AN ACFT X INTO TXK EARLIER IN THE SHIFT; SO I CALLED THE TXK ATCT AND ASKED IF THEY MIGHT BE WORKING A VFR ACFT X TO THE S OF THE FIELD. THEY REPLIED YES; AND THE ACFT IDENTED AS REQUESTED. I ASKED THE TWR IF THAT WAS THE SAME ACFT FROM EARLIER; AND HE REPLIED YES. IN REF TO THAT INCIDENT; I WORKED A TFC PATTERN INTO TXK RWY 22; WHERE THE SEQUENCE WAS #1/IFR ACR ACFT Z (L TFC); #2 VFR ACFT X (R TFC); AND #3 IFR GA (R TFC). THE ACFT X RPTED THE ARPT IN SIGHT; I ADVISED HIM HE WAS #2 TO FOLLOW AN ACFT Z L TFC; AND SWITCHED HIM TO TWR. I LATER OBSERVED THE VFR TARGET AND THE ACFT Z TARGET MERGE ON 4 MI FINAL. I THEN OBSERVED A BEACON ONLY TARGET EXECUTE A L 360 DEG TURN. LATER STILL; I ASKED THE TWR CTLR IF HE HAD PUT THE ACFT X IN FRONT OF THE ACFT Z; AND HE REPLIED 'NO; THE ACFT X CUT HIM OFF AND HAD TO GO AROUND.'