Generalizations of the Theorems of Apollonius and Euler

Abstract

We present an algebraic generalization of Euler's theorem for quadrilaterals. Starting from the parallelogram identity in an inner product space, we derive Apollonius' identity and obtain Euler's quadrilateral identity in a unified vector framework. Using combinatorial approach, we establish a general algebraic relation for $n\geq 4$ vectors in an arbitrary real or complex inner product space. This result shows that Euler's classical relation is a special case of a general identity involving sums of squared norms of vectors.

Figures (2)

• Figure 1: Euler's theorem in $\mathbb{R}^m$. The sum of the areas of the brown squares equals the combined areas of the blue squares together with four times the area of the green square. In general, these squares need not lie in the same plane, since the points $A,B,C,D$ are not necessarily coplanar.
• Figure 2: Generalized "parallelogram" identity \ref{['eq.3.3']} for $n=6$ in $\mathbb{R}^m$. If the centroids $G_1$ and $G_2$ of triangles $\triangle A_1A_3A_5$ and $\triangle A_2A_4A_6$, respectively, are equal, then the sum of the areas of the six brown squares equals the difference between the total area of the six green squares and that of the three blue squares. In general, the squares need not lie in the same plane, since the points $A_1,A_2,...,A_6$ are not necessarily coplanar.

Theorems & Definitions (30)

• Theorem 1.1: Apollonius
• Theorem 1.2: Euler 1748
• Theorem 1.3: Parallelogram Identity
• Example 1.1
• Theorem 1.4: Apollonius
• proof
• Theorem 1.5: Euler
• proof
• Theorem 2.1
• proof