A PA28 CHEROKEE PVT PLT WITH A CFI ONBOARD ON FINAL APCH FOR RWY 19R NARROWLY MISS ANOTHER PA28 ON SHORT FINAL TO RWY 19R AT DPA; IL.

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FLT ORIGINATED AT DPA AT XA00 AS A SIGHT-SEEING FLT AROUND CHICAGO. AS A 200 HR PVT PLT SEL; I CONTRACTED WITH LCL FBO FOR USE OF AN ARCHER AND A CFI. BEFORE STARTING ENG; I ESTABLISHED THAT CFI WAS PIC FOR THE FLT. I HAD NO PAPERS AND WAS NOT FULLY PREPARED TO ACT AS PIC FOR FLT IN UNFAMILIAR AREA. I HANDLED ALL CTLS FOR THE DURATION OF THE FLT AND CFI HANDLED RADIOS; GPS; AND COMS. UPON RETURN; INITIAL CALL TO DPA ATC WAS 7.5 MI NE. THE FREQ WAS FAIRLY BUSY PRIOR. ATC GAVE US L BASE RWY 15 RPT 2 MI. ANOTHER ACFT CALLED IN A FEW MINS LATER FROM THE N AND GIVEN SIMILAR INSTRUCTIONS. I DON'T RECALL THE SPECIFICS BUT REMEMBER THAT THIS TFC COULD BE A FACTOR AND SUSPECT THAT THIS WAS THE OTHER ACFT INVOLVED. WHEN RPTING 2 MI BASE FOR RWY 15; ATC CALLED US #2 FOR RWY 19R FOLLOWING ACFT ON R BASE. WE WERE BOTH WATCHING FOR TFC. I SPOTTED AN ACFT (Z) WHICH APPEARED TO BE ON R BASE A FEW HUNDRED FT AGL AND 1/4-1/2 MI OUT. IN HINDSIGHT; ACFT MAY HAVE BEEN ON FINAL FOR RWY 15. I POINTED OUT TFC TO CFI AND CONTINUED MY APCH AND SCAN FOR OTHER ACFT. APPROX 1 MI OUT; 1500 FT MSL; ON/NEAR FINAL; I SPOTTED AN ACFT 25 FT R; 10 FT HIGH; AND SLIGHTLY BEHIND. I PITCHED DOWN; BANKED L SLIGHTLY TO AVOID AS I POINTED OUT TFC TO CFI. WE CONTINUED INTO A 360 DEGS AS I BEGAN A CLB. CFI WAS INITIALLY UNABLE TO CONTACT ATC. I HEARD GAR FROM SOMEONE. ATC CALLED US 'ACFT PERFORMING 360 DEGS' (WHICH WAS PROMPTLY REPLIED) AND INSTRUCTED US TO DEPART TO W 1 MI AND RE-ENTER R BASE RWY 19R. I COMPLIED AND LANDED WITHOUT FURTHER INCIDENT. TWR CTLR INDICATED IN AIR (DURING 360 DEGS) THAT WE WERE TO FOLLOW THE OTHER INCIDENT ACFT -- NOT PARTICULARLY USEFUL INFO AT THE TIME. ALSO; TWR CTLR CHASTISED US FOR USE OF 360 DEGS BECAUSE HE HAD OTHER TFC WHEN HE SWITCHED TO GND. CFI RESPONDED WITH A FEW TERSE COMMENTS THAT WE HAD TO AVOID OTHER ACFT. CONTRIBUTING FACTORS: AS A PLT UNFAMILIAR WITH DPA TFC PATTERNS; I COULD EASILY PERCEIVE AN ACFT ON FINAL FOR RWY 15 AS A R BASE FOR RWY 19R. IF RADIO TFC WAS LESS AND I WAS HANDLING THE RADIO; I MAY HAVE BEEN INCLINED TO CALL NEGATIVE TFC AT SOME POINT. CFI COMMENTED BEFORE CHANGE OF RWYS THAT I WAS ON A 'PERFECT' L BASE FOR RWY 15. FROM THE POS I WAS GIVEN RWY 19; I WAS SET UP FOR A DOG-LEG TO FINAL WHICH MAY NOT HAVE PROVIDED THE BEST VISIBILITY TO SEE OTHER ACFT ESTABLISHED ON A LONG FINAL. RWY LAYOUT AT DPA IS NOT IDEAL FOR MULTIPLE LNDG AND TKOF USE. RWY (EXTENSIONS) CROSS IN THE AIR WHICH HAS TO BE MORE DANGEROUS THAN ON THE GND. I WOULD HAVE PREFERRED TO 'STACK UP' IN THE PATTERN OR USE A SHORTER RWY 19L OVER AN APCH TO RWY 15 WHICH CROSSES RWY 19R. ALSO; I UNDERSTAND THAT ATC HAS ARPT SURVEILLANCE RADAR. OBVIOUSLY; THIS WASN'T EFFECTIVE IN AVOIDING TFC CONFLICT. THERE WERE NO APPARENT TA'S.