A PA28 PVT PLT NARROWLY AVOIDS AN OVERTAKING; CLBING BE36 OFF FREQ WITH APCH CTL AND SKIRTING THE FRINGE OF CLASS C AIRSPACE NEAR BDL; CT.

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#1 HDG 070 DEGS; I THINK JUST OUTSIDE CLASS C AIRSPACE AT BDL ON 125.8 APCH/DEP. I DON'T KNOW IF THERE WAS A CONFLICT ALERT; BUT CTLR CALLED 'CONVERGING TFC AT 9 O'CLOCK POS' -- SOMETHING IN HIS VOICE! QUICK GLANCE OVER SHOULDER FULL BACK AND L ON YOKE RECOVERED AT 3800 FT. APPROX 3 SECONDS LATER; PLANE #2 BELOW AND AHEAD ON COURSE 020 DEGS (APPROX). AT THE SAME TIME ATC DIRECTED FREQUENCY CHANGE TO 123.95. I ALSO INFORMED ATC CTLR OF NMAC. AT THAT TIME; HE TOLD ME THAT THEY WERE NOT IN CONTACT WITH THAT PLANE. FROM READING I KNOW THAT I WAS CLOSE TO THE MOST LIKELY POS TO HAVE THIS HAPPEN -- VMC; GOOD VISIBILITY; 10 NM TO BUSY ARPT! USUALLY RADAR ADVISORIES GIVE MUCH MORE WARNING; BUT MANY TIMES I HAVE VISUAL BEFORE THE CALL. I DON'T KNOW WHERE #2 CAME FROM. I WOULD EXPECT TO HAVE SEEN OFF MY R WING. AT THE TIME THE SUN WAS OFF THE R WING. I DON'T THINK #2 CAME UP FROM BELOW; CLOSURE RATE WAS MUCH TOO GREAT. THE REAL MUSHROOM EFFECT. I WAS LIT UP -- STROBES; BEACON; AND RECOGNITION LIGHTS. THANKS TO THE CTLR I'M HERE. I'M SORRY TO SAY THAT I DON'T THINK I WOULD HAVE SPOTTED #2 ON MY OWN IN TIME. I ALSO DO NOT BELIEVE THAT #2 WOULD HAVE SPOTTED ME. I MUST HAVE BEEN BIG AS A HOUSE IN FRONT OF HIM WHEN I TOOK ACTION. I ESTIMATE 1-2 SECONDS FROM COLLISION. I ONLY WISH THE 'OTHER GUY' HAD A CLUE HOW CLOSE IT WAS. I DON'T BELIEVE 'HE' DOES. MY GUESS IS THE AUTOPLT WAS FLYING AND PLT WAS 'INSIDE THE COCKPIT.'