Cevian Nest: What Is It About?
A Mathematical Droodle
What if applet does not run? 
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Copyright © 19962017 Alexander Bogomolny
The applet may suggest the following statement:
Let ΔQ_{1}Q_{2}Q_{3} be the cevian triangle of point P with respect to ΔP_{1}P_{2}P_{3}. Let ΔR_{1}R_{2}R_{3} be the cevian triangle of point Q with respect to ΔQ_{1}Q_{2}Q_{3}. Then triangles P_{1}P_{2}P_{3} and R_{1}R_{2}R_{3} are perspective (from a point): the lines P_{1}R_{1}, P_{2}R_{2}, P_{3}R_{3} are concurrent.
( ΔQ_{1}Q_{2}Q_{3} is called a cevian triangle triangle of point P with respect to ΔP_{1}P_{2}P_{3}, if its vertices Q_{1}, Q_{2}, Q_{3} serve as the feet of the cevians P_{1}Q_{1}, P_{2}Q_{2}, P_{3}Q_{3} in ΔP_{1}P_{2}P_{3} through point P.)
The configuration of three triangles inscribed into each other is known as the Cevian Nest.
What if applet does not run? 
The problem has been proposed by H. Gülicher (#1581, Mathematics Magazine Vol. 72, No. 4, October 1999). A solution by Daniele Donini has been published a year later (Mathematics Magazine Vol. 73, No. 4, October 2000, p. 325).
All subscripts below are interpreted cyclically modulo 3, so that
 =  1 
and
 =  1 
By the invariance of crossratio of four collinear points under projection (from P_{k}),
 = 

for k = 1, 2, 3. The product of the three equalities simplifies to
 = 

These two expressions are therefore either both equal to 1 or both are not. By Ceva's theorem, the first is equal to one only if the lines P_{1}T_{1}, P_{2}T_{2}, P_{3}T_{3} are concurrent. The second is equal to 1 only if the lines Q_{1}R_{1}, Q_{2}R_{2}, Q_{3}R_{3} are concurrent.
Note: Vladimir Nikolin had a somewhat different perspective on Cevian nests.
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 Store
Copyright © 19962017 Alexander Bogomolny
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