Why Are the Gergonne and Soddy Lines Perpendicular? A Synthetic Approach

  In any scalene triangle the three points of tangency of the encircle together with the three vertices can be used to define three new points which are, remarkably, always collinear,and this line is called the Gergonne Line. Moreover cevians through these tangent points are always concurrent at a c...

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Bibliographic Details
Published in:Mathematics magazine 2008-06, Vol.81 (3), p.211-214
Main Author: Feng, Zuming
Format: Magazinearticle
Language:English
Subjects:
Circles
Geometry
Mathematical problems
Mathematical theorems
Numerical analysis
Perpendicular lines
Property lines
Tangents
Triangles
Trigonometry
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Summary:  In any scalene triangle the three points of tangency of the encircle together with the three vertices can be used to define three new points which are, remarkably, always collinear,and this line is called the Gergonne Line. Moreover cevians through these tangent points are always concurrent at a common point that, together with the incenter, defines a second line, the Soddy Line. Feng discusses why should these lines be perpendicular and uses the classical theorems of Ceva and Menelaus to define these lines and then establish their perpendicularity by using a certain inversion.
ISSN:0025-570X
1930-0980
DOI:10.1080/0025570X.2008.11953551