B737 FLT CREW DOES GAR FROM APPROACH TO RWY 25L AT LAS AND CONFLICTS WITH ACFT DEPARTING 25R CAUSING TCAS RA.

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ON APCH TO RWY 25L; ATC CLRED US DIRECT TO LARRE; MAINTAIN 8000 FT. WE WERE HELD AT 8000 FT UNTIL LARRE WHEN WE WERE CLRED FOR THE VISUAL AND TOLD TO MAINTAIN 190 KTS AND SWITCH TO TWR. I CHKED IN AND TWR CLRED US TO LAND AND TO MAINTAIN 170 KTS. I INFORMED THEM THAT WE WERE UNABLE THE AIRSPD BECAUSE WE NEEDED TO CONFIGURE AND GET DOWN. HE RELIEVED THE REQUIREMENT BUT BECAUSE WE WERE LEFT HIGH AND HAD STRONG TAILWINDS ON FINAL; WE WERE UNABLE TO GET STABILIZED; AND I DIRECTED THE FO TO GO AROUND. THE TWR DIRECTED US TO FLY RWY HDG AND CLB TO 5000 FT. HE ALSO POINTED OUT COMPANY TFC ROLLING FOR TKOF ON RWY 25R. TWR THEN DIRECTED US TO TURN L AT 3 DME TO HDG 200 DEGS. I ACKNOWLEDGED AND SWITCHED BACK TO THE VOR. WHILE ON THE GO; WE LOST SIGHT OF THE ACFT TAKING OFF FROM RWY 25R. (THE FOLLOWING OCCURRED ALMOST SIMULTANEOUSLY.) AT APPROX 1.5 DME; TWR ASKED US IF WE HAD THE ACFT IN SIGHT. I RESPONDED THAT WE DID NOT AND WE RECEIVED A TA. TCAS INDICATED THE ACFT AT 1 O'CLOCK POS; 800 FT LOW AND INSIDE 1 MI. TWR DIRECTED US TO TURN EARLY AND WE BEGAN THE TURN. IMMEDIATELY AFTER STARTING THE TURN; WE RECEIVED AN RA. I GAINED SIGHT OF THE ACFT; TOOK CTL FROM THE FO AND RESPONDED TO THE RA. I CANNOT DEFINITIVELY SAY; BUT I BELIEVE WE GOT TO WITHIN AT LEAST 500 FT OF EACH OTHER. AFTER THE CONFLICT WAS RESOLVED; I DSNDED BACK TO 5000 FT AND WE WERE SWITCHED TO APCH CTL FOR VECTORS BACK TO THE RWY 19L. I SUSPECT THE ACFT TAKING OFF FROM RWY 25R WAS ON AN RNAV DEP AND THUS WOULD BE FLYING A DEFINED TRACK. WE WERE ON AN ASSIGNED HDG WITH STRONG WINDS FROM THE SSE CAUSING US TO DRIFT TOWARDS RWY 25R. AFTER OUR GAR; TWR WAS IN THE PROCESS OF TURNING THE ARPT AROUND AND THEY GOT VERY BUSY. WHILE THE CTLR DID CATCH THE SITUATION IN TIME; IT DIDN'T NEED TO BE THAT CLOSE. THE CLRNC OF RWY HDG TO 3 DME ON A GAR WITH ANOTHER ACFT ROLLING ON THE PARALLEL IS NOT EFFECTIVE. HAD OUR TURN BEEN STARTED EARLIER; THIS SITUATION WOULD NOT HAVE OCCURRED. IN ADDITION; THOUGH WE WERE NOT DIRECTED TO; IT IS NEARLY IMPOSSIBLE TO KEEP THE OTHER AIRPLANE IN SIGHT WITH OVERTAKE AND US DRIFTING CLOSER TO HIS RWY. WITH NO WAY TO MAINTAIN VISUAL SEPARATION; IT IS UP TO ATC TO ENSURE THESE SITUATIONS DO NOT OCCUR.