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PROFESSOR KELLAND ON THE THEORY OF PARALLELS. 



Definition. The point of intersection of the two diagonals is the centre of the 

 parallelogram. 



Prop. XXVI. Parallelograms upon equal bases may he between the same parallels 



only when they have the same centre. 



For if other parallelograms were upon equal bases, the perpendiculars through 

 their centres on one of the parallels would also be perpendicular on the other 

 (Prop., XXV.) which is impossible (Ax. Cor. 2). 



Prop. XXVII. Upon a given diagonal to describe a square. 



Let AB be the given diagonal ; it is required to describe a square on AB as 

 diagonal. p 



Bisect AB in C ; draw CD perpendicular to AB, and pro- 

 duce it, and make DC, CE, each equal to AC. ADBE is the 

 square required. 



The triangles are all equal (Euc. I. 4). whence the con- 

 clusion is obvious. 



Cor. A square is a parallelogram. 





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