AN ACR FLC FLYING A B727 FROM ORD RPTS AN NMAC WITH ANOTHER ACR THAT WAS IN THE PROCESS OF A GAR.

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I WAS CAPT AND PF ON A B727 DEPARTING ORD RWY 14L. OUR ACFT WAS LIGHT WT AND WE HAD A MAINT DEFERRAL THAT REQUIRED A MAX PWR TKOF (AUTO PACK TRIP WAS INOP). DUE TO THE LIGHT WT; I SELECTED FLAPS 5 DEGS FOR TKOF. THESE FACTORS COMBINED TO GIVE US AN UNUSUALLY RAPID ACCELERATION AND VERY HIGH RATE OF CLB. JUST AFTER GEAR RETRACTION; ORD TWR ADVISED US THAT WE WOULD SEE AN ACFT Y GOING MISSED APCH OFF RWY 4R. BOTH THE FO AND I TRIED TO SEE THE OTHER ACFT (ITS A FLT PATH WAS FROM 100 DEGS TO OUR R) BUT IT WASN'T VISIBLE. WE THEN LOOKED AT THE TCASII DISPLAY AND I SAW A TARGET SLIGHTLY R AND READING ABOUT 200 FT BELOW US. I INCREASED THE PITCH ANGLE TO ABOUT 20 DEGS NOSE HIGH WITH THE ACFT STILL AT FLAPS 5 DEGS AND MAX PWR. AIRSPD WAS ABOUT 170 KIAS AND DECREASING SLOWLY. AS THE TARGET PASSED BELOW US; THE TCASII DISPLAYED 300 FT THEN 400 FT BELOW. SHORTLY AFTER THAT; I SAW A B757 OR B767 APPEAR BELOW US AND TO THE L. OBVIOUSLY; OUR LIGHT WT AND UNUSUALLY HIGH RATE OF CLB HELPED US AVOID AN NMAC. WE DID NOT GET A TCASII RA/TA. MY CONCERN IS THAT HAVING TKOFS AND ARRS ON RWYS WITH EXTENDED CTRLINES THAT CROSS IS VERY DANGEROUS IF A GAR OCCURS.