Cevian Nest: What Is It About?
A Mathematical Droodle
What if applet does not run? |
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Copyright © 1996-2018 Alexander Bogomolny
The applet may suggest the following statement:
Let ΔQ1Q2Q3 be the cevian triangle of point P with respect to ΔP1P2P3. Let ΔR1R2R3 be the cevian triangle of point Q with respect to ΔQ1Q2Q3. Then triangles P1P2P3 and R1R2R3 are perspective (from a point): the lines P1R1, P2R2, P3R3 are concurrent.
( ΔQ1Q2Q3 is called a cevian triangle triangle of point P with respect to ΔP1P2P3, if its vertices Q1, Q2, Q3 serve as the feet of the cevians P1Q1, P2Q2, P3Q3 in ΔP1P2P3 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 cross-ratio of four collinear points under projection (from Pk),
| = |
|
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 P1T1, P2T2, P3T3 are concurrent. The second is equal to 1 only if the lines Q1R1, Q2R2, Q3R3 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|
Copyright © 1996-2018 Alexander Bogomolny
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