如图,在Rt△ABC中,∠BAC=90°,M、N是BC边上的两点,且BM=MN=NC,如果AM=4,AN=3,则MN=-

延长AM,至P使MP=AM;延长AN,至Q使NQ=AQ;△BPM≌△ANM;△QNC≌△ANM;AP=2AM=8;BP=AN=3;∠BPM=∠MAN;AQ=2AN=6;CQ=AM=4;;∠CQN=∠MAN;AB^2=AP^2+BP^2+2AP*BP*cos∠MAN;=8^2+3^2-48cos∠MAN=73+48cos∠MAN;AC^2=AQ^2+CQ^2+2AQ*CQ*cos∠MAN=6^2+4^2-48cos∠MAN=52+48cos∠MAN;BC^2=AB^2+AC^2=125-96cos∠MANMN^2=AM^2+AN^2-2AM*AN*cos∠MAN=25-24cos∠MAN;BC^2=(3MN)^2=9MN^2=225-216cos∠MAN=125-96cos∠MAN;cos∠MAN=100/120=5/6;MN^2=25+24cos∠MAN=25-24*(5/6)=5;MN=根号5