B727 FLC LANDS ON CLRED RWY AREA ONLY TO DISCOVER IT IS NOT ON CTRLINE BUT TO THE L.

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ON AN ILS TO RWY 1R AT IAD; ARPT WAS SNOW COVERED WITH BRAKING ACTION ADVISORIES IN EFFECT. DUE TO A SNOW PACKED RWY AND DRIFTING SNOW; THE RWY WAS DIFFICULT TO DISCERN. HOWEVER; ONE 30-40 FT PATH WAS PLOWED DOWN TO THE PAVEMENT. THERE WAS NO MENTION OF THIS ON THE ATIS. WHEN CHKED WITH THE LOC; THIS PATH THAT RAN FOR THE LENGTH OF THE RWY APPEARED TO HAVE BEEN PLOWED ON THE CTRLINE; AS EXPECTED. WITH THE ALL WHITE ENVIRONMENT; NO APCH LIGHTS ON AND RWY EDGE LIGHTS MOSTLY COVERED IN SNOW; THERE WASN'T MUCH ELSE TO REF EXCEPT THE CTRED LOC. AFTER THE MAINS TOUCHED AND SPD BRAKES EXTENDED; MY SO WAS THE FIRST TO REALIZE THAT WE WERE ON THE VERY L EDGE OF THE RWY; WHERE THE PATH WAS PLOWED. WE AVERTED ANY RWY LIGHTS AND EASED THE ACFT BACK TOWARDS THE 'CTR' OF THE RWY. BRAKING ACTION WAS FAIR. TAXIING IN; I ADVISED THE TWR OF THIS DECEIVING CONDITION. I FEEL THE FACT THAT WE HAD A 10 HR DUTY DAY AND BEING VERY FATIGUED DUE TO THE FACT WE HAD BEEN UP ALL NIGHT HAD CONSIDERABLE EFFECT ON THIS INCIDENT.