NMAC IN TFC PATTERN AT RNT; WA.

Narrative

I WAS CLRED TO LAND ON RWY 15 AT RENTON; FOLLOWING A CESSNA ON FINAL. BEHIND ME WAS A TWIN COMANCHE. RNT TWR ASKED ME TO GO AROUND (TO ALLOW THE TWIN TO LAND FIRST); WHILE I WAS STILL ON FINAL; AND I COMPLIED. THEN RNT TWR TOLD THE TWIN COMANCHE TO ALSO GO AROUND; AS THE CESSNA DID NOT EXIT THE RWY IN TIME. RNT TWR THEN TOLD ME TO 'MAKE YOUR TURN TO DOWNWIND NOW;' WHILE I WAS ON UPWIND AT ABOUT MIDFIELD; SO I TURNED (WITHOUT FLYING A XWIND LEG) ONTO A CLOSE-IN DOWNWIND. RNT TWR ADVISED ME THAT I WOULD BE ABLE TO MAKE MY BASE TURN SOON. I PROCEEDED NBOUND TOWARDS THE E CHANNEL BRIDGE (BTWN BELLEVUE AND MERCER ISLAND; ON THE 343 DEG RADIAL; 5.3 NM FROM RNT NDB); OVER THE WATER; ON A CLOSE-IN EXTENDED DOWNWIND. I WAS EXPECTING RNT TWR TO TELL ME WHEN TO TURN BASE; BUT HE DID NOT. AROUND THE SAME TIME; AN AIRPLANE RPTED INBOUND AT THE E CHANNEL BRIDGE. I WAS LOOKING FOR THE TFC AND NOW ASSUME THAT RNT TWR WANTED ME BEHIND HIM ALSO. I SAW THE OTHER AIRPLANE AND ESTIMATED WE WOULD MISS BY A FEW HUNDRED FT AND THUS I WOULD NOT NEED TO ALTER MY COURSE; AND THEN I SAW HIM BANK ABOUT 15 DEGS TO HIS R; TO TURN AWAY FROM ME. THE NEW TFC WAS TOLD TO FOLLOW ME AND SOON IT BECAME EVIDENT THAT RNT TWR HAD WANTED ME TO TURN BASE WITHOUT HIM TELLING ME. I FOLLOWED THE NEW TFC TO AS INSTRUCTED BY RNT TWR AND WAS CLRED TO LAND. RNT TWR FREQ WAS VERY BUSY AND THERE WAS NO OPPORTUNITY TO ASK FOR CLARIFICATION. I WAS ADVISED THAT THE PLT OF THE AIRPLANE THAT I FOLLOWED TO LAND CONSIDERED THE SIT UNSAFE AT THE TIME AND TOOK EVASIVE ACTION; WHICH WAS THE BANKING OF THE AIRPLANE TO HIS R THAT I SAW.