OKLAHOMA ACADEMY OF SCIENCE 49 



ON THE I— CENTERS OF A TRIANGLE 



Nathan Altshiller Court, 1917. 

 (Abstract) 



The sides of a triangle are touched by four circles the centers 

 of which are referred to as the I-centers of the triangle. 



Each I-center of a triangle lies on three of the six circles 

 which pass through the pairs of vertices of the triangle and have 

 their centers at the midpoints of the arcs substended by the res- 

 pective sides of the triangle on its circumcircle. 



Each of the six circles contains two I-centers, and the two 

 points are the extremities of a diameter. 



If th»se propositions are considered in connection with the 

 four triangles obtained from an inscribed quadrilateral by omit- 

 ting in turn one of its four vertices, the following properties are 

 found : 



The sixteen I-centers of the four triangles obtained by tak- 

 ing the vertices of an inscribed quadrilateral three at a time, lie 

 on twelve circles, each I-center lies on three circles. Each circle 

 contains four I-centers. The centers of those twelve circles lie 

 on the circucircle of the quadrilateral. 



The four incenters of the four triangles determined by the 

 vertices of an inscribed quadrilateral taken three at a time, form 

 a rectangle. 



If of the four triangles determined by the vertices of an in- 

 scribed quadrilateral taken three at a time, the three triangles are 

 taken having a vertex in common, the three excenteis relative to 

 this vertex in the three triangles, are the vertices of a rectangle, 

 the fourth vertex of which is the incenter of the fourth triangle. 



The sixteen I-centers of the four triangles determined by the 

 vertices of an inscribed quadrilateral taken three at a time, lie by 

 groups of four on eight straight lines. These eight lines consist of 

 two sets of four parallel lines, and the lines of one set are perpen- 

 dicular to the lines of the other. 



*The paper has appeared in full in the American Mathematical 

 Mnr^rhlv. vol. XXV, pp.241 -246, June, 1918. 



