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Generalization of Ceva theorem

Wojciech Wdowski

Abstract

In this paper, we present a novel generalization of the classical Ceva theorem to arbitrarily dimensional simplexes. Our approach allows cevians to have any dimension (smaller than the dimension of the base simplex). Consequently, our result unifies other generalizations of the Ceva theorem obtained in recent years.

Generalization of Ceva theorem

Abstract

In this paper, we present a novel generalization of the classical Ceva theorem to arbitrarily dimensional simplexes. Our approach allows cevians to have any dimension (smaller than the dimension of the base simplex). Consequently, our result unifies other generalizations of the Ceva theorem obtained in recent years.

Paper Structure

This paper contains 9 sections, 5 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Ceva's Theorem
  3. Simplices
  4. Construction of $k$-dimension cevians
  5. l-Faces
  6. Induced cevians
  7. Multipedes
  8. Main results
  9. Future work

Key Result

Theorem 1.1

Let $\Delta P_0P_1P_2$ be a triangle, and let points $Q_0,Q_1,Q_2$ lie on the edges opposite to vertices $P_0,P_1,P_2$, respectively (see Figure fig:trojkat_punkty_Q). Then, simplices $P_iQ_{i}$(for $i\in\{0,1,2\}$) intersect nonempty if and only if the following formula holds

Figures (3)

  • Figure 1: Demonstration of Ceva's Theorem.
  • Figure 2: Case $n=3, k=2$ with the $2\hbox{-cevian} C$ is dashed and the $2\hbox{-face} \mathbf{L}$ is dotted.
  • Figure 3: Visualization for Case $n=4, k=2$. The $2\hbox{-cevian} C$ is dashed, and the $3\hbox{-face} \mathbf{L}$ is dotted.

Theorems & Definitions (13)

  • Theorem 1.1: Ceva's Theorem
  • Example 1.2: Case $n=3\text{ and }k=2$
  • Example 1.3: Case $n=4$, $k=2$
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • proof : Proof of Theorem \ref{['main']}
  • Theorem 2.5
  • ...and 3 more