85 



Hamiuoud and other tiourisbing- cities in the iiuitbwest part of the 

 State would experience the most direct benetits by the increase of business 

 and manufacturing facilities and consequent increase in population. 



The proposition is Avorth at least a passing thought and is commendHd 

 to the State and general go\ernnionts for further consideratio!). 



Some Properties of the Sym.median Point. 



By Kobert J. Aley. 



Monsieur Emile Lemoine, at the Lj'ons meeting of the Frencli Associ- 

 ation for the advancement of tlie Sciences in 187:5, called attention to 

 a particular point within the triangle, which he called the center of anti- 

 parallel medians. Since that time a number of mathematicians have 

 studied the point and have discovered many of its properties. The point 

 is such an interesting one that a brief collection of its more sti-ikiug 

 properties may be of some value. No claim is made to completeness. 



DEFINITIONS OF THE POINT. 



1. The point of concurrency of the bisectors of all lines antiparallel 

 to the sides of the triangle. 



2. The point of concurrency of the lines isogonal conjugate to the 

 medians of the triangle; that is, the point of concurrency of the sym- 

 medians of the triangle. 



3. The point within the triangle, the sum of the scpiares of whose 

 distances from the three sides is the least possible. 



4. The point within the triangle, whose distances from tlie sides is 

 directly proportional to the sides. 



NAMES OF THE POINT. 



1. Center of antiparallel medians, proposed by Monsieur Emile Le- 

 moine. 



2. Symmedian point (symediane, from symetrique de la mediane), 

 proposed by Monsieur Maurice d'Ocagne. The Engish form "symmedian" 

 was suggested by Mr. R. Tucker in 1884. 



