甲、乙两人同解方程组Ax+By=2,Cx-3=-2,由于甲正确解得方程组x=1,y=-1,由于乙看错了C解得方程组得x=2,y=-6如图,DC和BE相交于点A,EF平分∠DEA,CF平分∠ACB.求证：∠F=2\1(∠B+∠D）.

Question

甲、乙两人同解方程组Ax+By=2,Cx-3=-2,由于甲正确解得方程组x=1,y=-1,由于乙看错了C解得方程组得x=2,y=-6
如图,DC和BE相交于点A,EF平分∠DEA,CF平分∠ACB.求证：∠F=2\1(∠B+∠D）.

Answer 1

甲正确解方程,故可将 x=1,y=-1 代入 Ax+By = 2,Cx-3 = -2,
可得：A-B = 2 ,C-3 = -2 ；解得：C = 1 ；
乙只看错了C,不影响第一个方程,故可将 x=2,y=-6 代入 Ax+By = 2,
可得：2A-6B = 2 ；
联立 A-B = 2 ,2A-6B = 2 ,解得：A = 5/2 ,B = 1/2 ；
综上可得：A = 5/2 ,B = 1/2 ,C = 1 .
∠F
= 180°-(∠FCE+∠FEC)
= 180°-(∠FCA+∠ACE+∠AEC+∠FEA)
= 180°-(∠ACE+∠AEC)-(∠FCA+∠FEA)
= ∠CAE-(1/2)(∠BCD+∠AED)
= (1/2)[2∠CAE-(∠BCD+∠AED)]
= (1/2)[(∠CAE-∠BCD)+(∠CAE-∠AED)]
= (1/2)(∠B+∠D)