因为AA1/A1B = BB1/B1C
所以AA1/BB1 = A1B/B1C = （AA1+A1B）/（BB1+B1C）(合比定理)
因为是正方形ABCD且AA1+A1B=AB BB1+B1C=BC
所以（AA1+A1B）/（BB1+B1C）=AA1/BB1 = A1B/B1C =1
所以AA1=BB1,A1B=B1C
同理可得AA1=BB1=CC1=DD1,A1B=B1C=C1D=D1A
且∠A=∠B=∠C=∠D
所以△AA1D1≌△A1BB1≌△B1CC1≌△C1DD1
所以A1B1=B1C1=C1D1=D1A1
所以四边形A1B1C1D1是菱形
因为∠BA1B1+∠BB1A1=90°
又因为∠BA1B1=∠C1B1C
所以∠C1B1C+∠BB1A1=90°
所以∠A1B1C1=90°
所以菱形A1B1C1D1是正方形.