86 



3. Minimum point, sufjf^ested by Dr. E. W. Grebe. 



4. Grebe's point, pioposed by Dr. A. Emmericli. 



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, making the same angle with the bisectors of angles A and B, respectively, 

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

 median point. 



2. Draw anti parallels 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 iymmedian 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, CZ concur at K, the 

 gymmedian 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 



KKa 



2Aa 1 



+ b2 + c 



KKb^= — y Where A is the area of the triangle 



• a^ -)- b^ -f- c^ I 



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



2 A c 

 KKc — 2 I K2 I 2 the same triangle. 



A q2 



Area of \ BKC 



Area of A CKA = 

 Area of A AKB 



a2 + b2 + c2 



Ab^ 

 a2 + b2 + c2 



Ac^ 

 a2 + b2 + c2 



A BKC : A CKA : A AKB = a^ : b^ ; c^ 



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. 



