86 



3. Minimum pomt, suggested by Dr. E. W. Grebe. 



4. Grebe's point, proposed by Dr. A. Emmerich. 



5. Lemoine's point, proposed by Professor J. Neuberg. 



METHODS OF CONSTRUCTING THE POINT. 



1. Draw the medians AMa, BMb of the triangle ABC. Then draw AK'a, 

 BK^b, maliing the same angle with the bisectors of angles A and B, respectively, 

 as are made by AMa and BMb. The intersection of AK^a, BK^b is K, the sym- 

 median point. 



2. Draw antiparallels to BC and CA. Join A and B, respectively, to the 

 midpoints of these antiparallels, and the intersection of these joining lines is K, 

 the Bymmedian point. 



3. To the circumcircle of the triangle draw tangents at B, C and A, and let 

 these intersect in X, Y, Z, respectively. Then AX, BY, GZ concur at K, the 

 aymmedian point. 



SOME PROPERTIES OF THE POINT. 



1. K is the point isogonal conjugate to G, the centroid. 



2. If Ka, Kb, Kc are the feet of the perpendiculars from K to the three sides 

 respectively, then 



2 Aa 1 



KKa 



KKb 



i2 + b2 + C2 



^ y Where A is the area of the triangle 



I ABC, and a, b, c are three sides of 



2 A c 

 KKc =^ 2 _|_ ua I 2 ^^^ same triangle. 



Area of A BKC = 

 Area of A CKA = 



Area of A AKB 



a2 + b2 4- c2 



Ab'' 

 a2 + b2 + c2 



Ac^ 

 a2 + b2 -f c2 



A BKC : A CKA : A AKB = a^ : b^ : cK 



4. Antiparallels to sides of the triangle through K are equal. Such anti- 

 parallels cut the sides of the triangle in six points which lie on a circle whose 

 centre is K. This circle is called the Cosine Circle. 



5. K is the median point of the triangle KaKbKc. 



