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COLLINEAR SEts OF THREE Points CONNECTED WITH THE TRIANGLE. By 
Ropert J. ALEY. 
This paper does not claim to be either original or complete. It contains a 
fairly complete list of collinear sets connected with the triangle. All cases of 
collinearity connected with polygons of more than three sides have been omitted. 
The subject of collinearity is both interesting and fruitful. There are three 
well defined methods of proving the collinearity of three points. The classic one 
is the application of the theorem of Menelaus: ‘‘If D, E, F are points on the 
sides BC, C A, AB respectfully of AB C, such that B DxCEXA F=— DCXE 
AF B, then D, E, F are collinear.” In many cases the data are insufficient for 
the use of this method. Another method of frequent use is to prove that the angle 
formed by the three points is a straight angle. The author has used another 
method, believed to be original with him, when the points in question are such 
that the ratios of their distances from the sides can be determined. This method 
is fully illustrated in ‘‘ Contributions to the Geometry of the Triangle.” 
Collinear problems fall into two very well marked classes. The first class is 
made up of those points which are definitely located with respect to the triangle, 
The second class is made up of those points which are located with reference to 
some auxiliary point. 
NOTATION. 
In order to save time in the enunciation of propositions the following notation 
will be used: 
A B C is the fundamental triangle. 
A, B, ©, is Brocard’s first triangle. 
Ma My M¢ is the triangle formed by joining the middle points of the sides of 
ABC. 
Mia Mi» Mie is the triangle formed by joining the middle points of the sides 
of A, By:Cy. 
M is the centre of the circumcircle of AB C. 
M! is the point isotomic conjugate to M. 
G is the median point or Centroid of AB C. 
K is Grebe’s point or the Symmedian point. 
D is the centre of perspective of AB C and A, B, C,. 
D! is the point isogonal conjugate to D. 
H is the Orthocentre. 
@ and 2! are the two Brocard points. 
N is Tarry’s point. 
