AN ATX BE58 PLT HAS A NMAC WITH AN UNIDENTED VFR ACFT. THE APCH CTLR WAS UNABLE TO TRACK THE OTHER ACFT.

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I WAS LEVEL AT 5000 FT ON A 210 DEG HDG AS ASSIGNED BY ICT APCH. ICT APCH CALLED OUT TFC AT 11-12 O'CLOCK POS; 4 MI CLBING THROUGH 4700 FT MSL. THEY THEN ISSUED ME TO EXECUTE A TURN 10 DEGS MORE TO THE R. AS I LEVELED OFF WE CALL THAT WE HAD THE TFC IN SIGHT. SAW THAT THE CLOSURE RATE WAS INCREASING AND THAT A MIDAIR WAS INEVITABLE. I MADE A R HAND TURN IN A 50-60 DEG BANK; 45 DEG HDG CHANGE. DURING THE MIDDLE OF THE TURN I WITNESS THE ACFT PASS BTWN THE NACELLE AND THE FUSELAGE AT APPROX 200 FT. THE COLOR OF THE ACFT WAS BLUE; SILVER; AND WHITE. IT WAS EITHER A 58 BARON; OR A 36 BONNAZA. AFTER SECONDS THE ACFT CONTINUED STRAIGHT AHEAD. I ASSUME THEY NEVER SEEN ME. IT WOULD HAVE BEEN NICE TO HAVE A TFC A LITTLE FURTHER OUT THAN 4 MI. AFTER I WAS SURE THAT THE ACFT WAS CLR; I RETURNED TO MY ORIGINAL HDG OF 220 DEGS AND NOTIFIED ATC THAT THIS WAS A NEAR MISS. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR WAS THE PLT OF A BEECH 58 BARON ATX ON A CHARTERED BUSINESS FLT. HE SAID THAT HE WAS UNSURE OF THE OTHER ACFT'S TYPE BECAUSE HE ONLY SAW THE OTHER ACFT'S TAIL SECTION AS HE MANEUVERED TO MISS IT. ALTHOUGH IT WAS VFR THERE WAS SOME HAZE IN THE AREA. THE PLT SAID THAT THE APCH CTL REPRESENTATIVES HAVE CALLED HIM SEVERAL TIMES ABOUT THE INCIDENT AND HE ALLEGES THAT THEY TOLD HIM THAT THE ACFT WAS POP-UP TFC FROM A FIELD NEAR ICT THAT THEY WERE ABLE TO TRACK TO A POINT NEAR SLN; BUT THEN THEY LOST HIM AND HE DID NOT LAND AT SLN OR ANY OF THE ARPTS NEAR THERE.