A BRASILIA FLC WAS UNABLE TO MAINTAIN VISUAL SEPARATION AFTER ACCEPTING THE SAME ALT AS RPTED TFC NEAR SHV AND RECEIVED A TCASII RA.

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AROUND (STILL DARK); WHILE CRUISING AT 15000 FT ON A 230 DEG HDG; 40 NM SW OF SHV; ZFW CALLED TFC TO US AT 10 O'CLOCK POS (10-15 MI; BUT I'M NOT SURE) AND 16000 FT WBOUND. WE CALLED THE OTHER ACFT IN SIGHT AND WERE TOLD TO CLB TO 16000 FT; THE SAME ALT; AND MAINTAIN VISUAL SEPARATION. AFTER SEVERAL MINS AT THE SAME ALT ON CONVERGING HEADINGS; WE NOTICED VISUALLY AND ON THE TCASII THAT THE OTHER ACFT WAS GETTING A LITTLE CLOSE AND SOMETHING SHOULD BE DONE SOON. I IMMEDIATELY CALLED CTR AND ASKED IF THEY INTENDED TO GIVE US A VECTOR AROUND THE OTHER ACFT TO WHICH THE CTLR RESPONDED '(CALL SIGN) YOU WERE TOLD TO MAINTAIN VISUAL SEPARATION.' WE IMMEDIATELY BEGAN A TURN TO THE R; AWAY FROM THE OTHER ACFT. AT APPROX THAT SAME TIME WE RECEIVED A TCASII TA FOLLOWED BY AN RA TO CLB. THE SAAB SIMULTANEOUSLY ANNOUNCED THAT THEY WERE DSNDING FOR A TCASII RA. WE FLEW DIRECTLY OVER THE TOP OF THE OTHER ACFT AT APPROX 16300 FT WHILE THE OTHER ACFT ANNOUNCED THAT THEY HAD DSNDED 300 FT LOW FOR A TCASII RA. OUR MANEUVERING WAS VERY BENIGN SINCE WE SAW THE SIT DEVELOPING AND TOOK APPROPRIATE ACTION EARLY. I CAN ONLY ASSUME THE SAME FOR THE OTHER ACFT. ALTHOUGH THE ONUS WAS ON US TO MAINTAIN VISUAL SEPARATION IT WAS DIFFICULT TO JUDGE THE OTHER ACFT'S COURSE AND CLOSURE IN THE DARK; AS IT INITIALLY APPEARED TO BE MORE PARALLELING OUR COURSE. I BELIEVE MUCH OF THIS WAS ATTRIBUTED TO THE DARKNESS AND SEEING ONLY BLINKING LIGHTS RATHER THAN AN ASPECT ANGLE ON THE OTHER PLANE. WE EVENTUALLY REALIZED THAT OUR FLT PATHS WERE GOING TO BE TOO CLOSE FOR COMFORT; BEGAN A TURN; AND WERE BACKED UP BY THE TCASII. IN RETROSPECT; I BELIEVE THAT CTLRS SHOULD BE VERY WARY OF ASSIGNING 2 ACFT; WHICH ARE UNDER THEIR CTL AND ON CONVERGING COURSES; TO THE SAME ALT DURING HRS OF DARKNESS EVEN THOUGH INSTRUCTIONS ARE GIVEN TO MAINTAIN VISUAL SEPARATION. DARKNESS CAN EASILY PLAY TRICKS ON OUR PERCEPTIONS. IF WE HAD BEEN ASSIGNED SEPARATE ALTS OR A CORRECTIVE VECTOR; THE ENTIRE SIT COULD HAVE BEEN EASILY AVOIDED. TCASII PLAYED A CRUCIAL ROLE IN DIRECTING BOTH ACFT TO MANEUVER VERTICALLY IN OPPOSITE DIRECTIONS.