AN INSTRUCTOR PLT FLYING AN SMA HAD TO TAKE EVASIVE ACTION TAXI FAST TO CLR THE TKOF RUN OF A BIPLANE. MULTIPLE RWY OP PARALLEL RWYS.

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MY STUDENT AND I WERE RETURNING FROM AN INSTRUCTIONAL FLT. WE DETERMINED; THROUGH AWOS AND CTAF; THAT THE ACTIVE RWY WAS RWY 34. WE ENTERED THE PATTERN NORMALLY ON THE 45 DEG ENTRY TO DOWNWIND. AT ARLINGTON; FOR RWY 34; THERE ARE ACTUALLY 3 RWYS. 2 ARE GRASS; AND ARE NOT PUBLISHED; OTHER THAN IN THE LCL ARPT PROCS. THEY ARE SEPARATED BY A DITCH. THE E GRASS RWY; CLOSEST TO THE TXWY; IS RESERVED FOR GLIDERS. THE W GRASS RWY; CLOSEST TO THE PAVED RWY; IS USED BY PWRED ACFT. SINCE IT IS LOCATED BTWN 2 TXWYS (A1 AND A2); THE LENGTH OF THE W GRASS RWY IS ABOUT 1500 FT. MOST LCL PLTS ARE AWARE OF THESE GRASS RWYS. INITIALLY; I PLANNED ON LNDG ON THE GRASS RWY. HOWEVER; AS WE TURNED BASE; I NOTICED AN AIRPLANE; VEHICLES AND SOME PEOPLE ON THE GRASS RWY NEAR TXWY A2. THE AIRPLANE WAS A VERY RARE BIPLANE. IT IS OPERATED BY A MUSEUM. SO I CHANGED PLANS AND DECIDED TO LAND ON THE PAVED RWY. AS WE TURNED FINAL; I NOTICED THE BIPLANE BACK-TAXIING ON THE GRASS RWY. AS THIS IS UNUSUAL (TYPICALLY PEOPLE WILL USE THE TXWY AND ALMOST NEVER BACK-TAXI); I TOLD MY STUDENT TO BE READY FOR A GAR. THERE WAS ANOTHER AIRPLANE FOLLOWING US IN THE PATTERN AS WELL. WE LANDED NORMALLY; AND TURNED OFF THE PAVED RWY AT A2. AS WE TAXIED CLR OF THE RWY; WE SLOWED DOWN; DUE TO THE PEOPLE NEAR THE TXWY. HOWEVER; THE PEOPLE WERE CLR OF THE TXWY AND THERE WAS NO COLLISION HAZARD. AS WE TURNED OFF THE RWY; I TURNED MY HEAD TO CHK ON THE FOLLOWING AIRPLANE. WHAT GOT MY ATTN WAS THE BIPLANE ON A TKOF ROLL; HDG RIGHT FOR US. I IMMEDIATELY ADDED PWR AND GOT CLR OF THE RWYS. THE BIPLANE DID NOT PERFORM ANY EVASIVE ACTIONS. IN FACT; IT ACTUALLY SLIPPED TOWARDS US (AND THE PEOPLE); IN A 'SHOW PASS' WITHIN 5 FT OF THE GND. IT PASSED WITHIN 30 FT OF OUR TAIL AS IT WENT BY. IF I HAD NOT ADDED PWR AND EXPEDITED CLRING THE GRASS RWY; HE WOULD HAVE HIT US. WE WERE LNDG TFC. MY UNDERSTANDING OF THE REGS IS THAT LNDG; ACFT HAVE THE RIGHT-OF-WAY. HOWEVER; YOU CANNOT FORCE AN ACFT THAT HAS PREVIOUSLY LANDED OFF THE RWY. THE BIPLANE MUST HAVE STARTED HIS TKOF ROLL AFTER WE PASSED HIM ON OUR FLARE OR IN OUR LNDG ROLL. IN ANY CASE; HE OPERATED WITHOUT REGARD FOR ANY OTHER TFC. TYPICALLY PEOPLE WILL NOT DEPART THE GRASS RWY UNLESS OTHER TFC IS CLR. LATER; I OBSERVED THE SAME BIPLANE DEPARTING DOWNWIND INTO TFC ON FINAL; BOTH FOR THE PAVED RWY AND A GLIDER FOR THE GLIDER RWY. THE BIPLANE PASSED UNDER THE GLIDER. BEFORE TURNING FINAL; I NOTICED THE BIPLANE'S OPS WERE 'UNUSUAL.' THIS ALERTED ME TO WATCH IT A LITTLE MORE CLOSELY. HOWEVER; DURING THE LNDG; I LOST TRACK OF IT. IF AN AIRPLANE IS OPERATING 'UNUSUAL' AND POSSIBLY IN A DANGEROUS MANNER; I LEARNED NOT TO LOSE SIGHT OF IT.