93 
Take a circle whose diameter is thirty-six units; Angel’s method makes 
the side of a 36-gon equal to 3.33982 units, while the true length is 3.15776 
units. The larger number of sides makes the error of the method more 
apparent. 
ConcuURRENT SETS OF THREE LINES CONNECTED WITH THE TRIANGLE. 
By Rogpert Jupson ALEY. 
To the student of the pure geometry of the triangle, few subjects are more 
interesting than the concurrency of lines. The following collection of concur- 
rent sets of three lines has been made in the hope that it may prove of value to 
geometric students. No claim is made to completeness. The list is as complete 
as the author could make it with the material to which he had access. Many of 
the notes, and a large number of the propositions have been taken from the pub- 
lished papers of Dr. J. S. Mackay, of Edinburgh, perhaps the foremost student of 
the geometry of the triangle. No classification of the propositions seems possible 
and so none has been attempted. 
1. The median lines of a triangle are concurrent. The point of concur- 
rency, usually denoted by G, is called the median point or centroid. 
2. The in-symmedian lines of a triangle are concurrent. The point of con- 
currency is called the symmedian point or Grebe’s point, and is generally denoted 
by K. (Fora history of this point, see J. S. Mackay, in Proceedings of Edin- 
burgh Mathematical Society, Vol. XI.) 
3. The altitudes of a triangle are concurrent. The point of concurrency, 
usually denoted by H, is called the ortho centre. (This proposition occurs in 
Archimedes’s Lemmas and in Pappus’s Mathematical Collection. ) 
4, The internal angle bisectors of a triangle are concurrent. The point of 
concurrency is the center of the inscribed circle and is usually denoted by J. 
(Euclid LV, 4.) 
5. The internal bisector of any angle of a triangle and the external bisectors 
of the other two angles of the triangle are concurrent. The points of concur- 
rency, denoted by I,, J,, I, are the centers of the three escribed circles. 
6. The perpendiculars to the sides of a triangle at the midpoints of the 
sides concur at the center of the circumscribed circle. This point of concurrence is 
usually denoted by O. (Euclid.) 
7. Lines drawn from the vertices to the points of contact of the in-circle 
with the opposite sides are concurrent. (The point of concurrency, I’, is called 
the Gergonne Point. It was named by J. Neuberg after J. D. Gergonne.) 
