Mark Shattuck, Steiner-Lehmus type results related to the Gergonne point of a triangle,
Forum Geometricorum, 17 (2017) 49--62.
Abstract. A Gergonne cevian (abbreviated GC) is a line segment joining the vertex of a triangle with the point of tangency of the triangle's incircle with the opposite side. In this paper, we determine monotonicity results for various segments determined by the intersection of Gergonne cevians and angle bisectors (both internal and external) within a triangle. We first consider the problem, in response to a prior question, of comparing certain segment lengths determined by the intersection of a fixed GC with the external bisectors of the other two angles of a triangle. We then consider the analogous problem wherein one fixes an angle bisector and considers the segment lengths determined by its intersection with the GC's emanating from the other two vertices. Finally, we prove some results for a related question comparing segments determined by the intersection of the GC from angle B with an angle bisector from angle C within a triangle ABC to those determined by the intersection of a bisector from angle B with the GC from angle C.
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