Cevian Cradle: What Is It About?
A Mathematical Droodle
What if applet does not run? 
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
The applet may suggest the following statement:
Given ΔABC and points X, Y, Z on BC, CA, and AB. O is an arbitrary point. D is the intersection of OA and YZ,
What if applet does not run? 
In fact more is true, viz.,
(*) 

Note that this is true regardless of the position of point O that does not enter the identity at all.
Thus when one of the expression equals 1 (Ceva's condition), so is the other. This relation has been established earlier at the Cevian Nest page. Here I shall offer another derivation.
The problem as it was formulated above, has been posted in the January 1961 Mathematics Magazine with several ad hoc solutions in the September 1961 issue. What follows is a comment by D. Moody Bailey published in the MarchApril 1962 issue.
Drop the perpendiculars YG and ZH from Y and Z onto AO. Right triangles DZH and DYG are similar, implying
 · 

and, in a similar fashion,
 · 
 and 
 · 

Consequently,
 · 
 ) 
However, the expression in the parentheses equals 1 due to the Trigonometric form of Ceva's theorem applied in ΔABC and the Cevians through point O. Thus (*) holds precisely because the three cevians AO, BO, CO meet in a point (O) so that, perhaps less surprisingly now, it holds regardless of the specific location of O.
Menelaus and Ceva
 The Menelaus Theorem
 Menelaus Theorem: proofs ugly and elegant  A. Einstein's view
 Ceva's Theorem
 Ceva in Circumscribed Quadrilateral
 Ceva's Theorem: A Matter of Appreciation
 Ceva and Menelaus Meet on the Roads
 Menelaus From Ceva
 Menelaus and Ceva Theorems
 Ceva and Menelaus Theorems for Angle Bisectors
 Ceva's Theorem: Proof Without Words
 Cevian Cradle
 Cevian Cradle II
 Cevian Nest
 Cevian Triangle
 An Application of Ceva's Theorem
 Trigonometric Form of Ceva's Theorem
 Two Proofs of Menelaus Theorem
 Simultaneous Generalization of the Theorems of Ceva and Menelaus
 Menelaus from 3D
 Terquem's Theorem
 Cross Points in a Polygon
 Two Cevians and Proportions in a Triangle, II
 Concurrence Not from School Geometry
 Two Triangles Inscribed in a Conic  with Elementary Solution
 From One Collinearity to Another
 Concurrence in Right Triangle
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
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