NAPOLEON’S QUASIGROUPS

VEDRAN KRČADINAC

Abstract. Napoleon’s quasigroups are idempotent medial quasigroups satisfying the identity (ab · b)(b · ba) = b. In works by V.Volenec geometric terminology has been introduced in medial quasigroups, enabling proofs of many theorems of plane geometry to be carried out by formal calculations in a quasigroup. This class of quasigroups is particularly suited for proving Napoleon’s theorem and other similar theorems about equilateral triangles and centroids.

1. Introduction

Consider the Euclidean plane E^{2} and define multiplication of its points by a · b = c, where c is the centroid of the positively oriented equilateral triangle over \overline{ab}. The groupoid (E^{2}, ·) is an example of the so-called Napoleon’s quasigroups.

Definition 1.1. An idempotent medial quasigroup (Q, ·) is called a Napoleon’s quasigroup if the following identity holds:

(ab · b)(b · ba) = b.  (1)

This means that (Q, ·) is a uniquely left and right solvable groupoid, i.e. for every a, b ∈ Q there are unique x, y ∈ Q such that ax = b and ya = b hold (denoted by x = a \ b and y = b / a). Furthermore, (Q, ·) satisfies the identities of idempotency and mediality:

a · a = a,  (2)
ab · cd = ac · bd.  (3)

Immediate consequences are the identities known as elasticity, left and right distributivity:

ab · a = a · ba,  (4)
a · bc = ab · ac,  (5)
ab · c = ac · bc.  (6)

Date: February 11, 2010.

2000 Mathematics Subject Classification. 20N05.

Key words and phrases. Medial quasigroup, Napoleon’s theorem.

Figure 1. Geometric interpretation of identity (1).

The operation · is also left and right distributive over \ and /, e.g.

(a \ b)c = ac \ bc.  (7)

Regarding our introductory example, the axioms are most easily checked by using complex coordinates in the plane. By identifying E^{2} ≡ ℂ, the binary operation can be written as a·b = (1−q)a+qb, for q = \frac{1}{2} + \frac{i\sqrt{3}}{6}. This is evidently an idempotent medial quasigroup, and identity (1) follows from 3q^{2} − 3q + 1 = 0. We could also take a · b to be the centroid of the negatively oriented equilateral triangle over \overline{ab}, which would correspond to choosing the other root q = \frac{1}{2} − \frac{i\sqrt{3}}{6}.

A more general example is obtained by taking an Abelian group (Q, +) with an automorphism φ such that 3φ^{2}(x)−3φ(x)+x = 0, for all x ∈ Q, and defining a new binary operation by a·b = a+φ(b−a). This operation is obviously idempotent, medial and uniquely right solvable. The equation for φ can be written as 1_{Q} = 3φ∘(1_{Q}−φ). Hence, 1_{Q}−φ is a bijection and the operation is uniquely left solvable. Identity (1) also follows directly from the equation.

As a consequence of Toyoda’s representation theorem [7], this is in fact the most general example of Napoleon’s quasigroups.

Theorem 1.2. For every Napoleon’s quasigroup (Q, ·) there is an Abelian group (Q, +) with an automorphism φ such that 3φ^{2} − 3φ + 1_{Q} = 0 and a · b = a + φ(b − a), for all a, b ∈ Q.

Proof. According to a special version of Toyoda’s theorem for idempotent medial quasigroups, there is a commutative group (Q, +) with automorphism φ such that a·b = a+φ(b−a). Identity (1) is equivalent with the property 3φ^{2} − 3φ + 1_{Q} = 0, which is easily verified by direct computation. □

This theorem completely describes the structure of Napoleon’s quasigroups and reduces them to the study of Abelian groups with a special type of automorphism. Our main motivation is to use these quasigroups as a language for proving Napoleon’s theorem and some related theorems of plane geometry. It turns out that all necessary ingredients are encoded in the properties of a single binary operation, most importantly identity (1).

In the next section, geometric concepts such as equilateral triangles and midpoints are defined in Napoleon’s quasigroups. We fall back to Toyoda’s representation theorem to prove some of the more technical results in this section. These results could also be proved directly, by somewhat tedious calculations in the quasigroup.

A surprisingly large number of results related to Napoleon’s theorem have been published over the years. A survey up to 1996 can be found in [5], and Napoleon-like theorems have kept appearing since. In the third section we prove Napoleons’s theorem and a well-known fact about centroids in the general context of Napoleon’s quasigroups. The solution to an old problem by E. Lemoine [4] is provided in this context. Finally, two more recent theorems by B. Grünbaum [1] and F. van Lamoen [3] are stated and proved in Napoleon’s quasigroups.

The challenge here lies not so much in the proofs, but in how these results should be formulated in the more general context. As illustrated by Theorem 3.6, a literal translation is not always correct. Once the result is formulated correctly, the proof in the quasigroup context is usually fairly straightforward.

This approach could make geometric theorems accessible to automated theorem provers. We do not pursue it in this work, but the use of automated theorem provers has become quite widespread in quasigroup and loop theory (see section 3 of [6] for a catalogue of results obtained in this way). The method of translating problems in plane geometry to the language of medial quasigroups is due to V. Volenec [8, 9]. Another class of idempotent medial quasigroup related to Napoleon’s are the hexagonal quasigroups [10]. As we shall see in the third section, a special hexagonal quasigroup can be obtained from an arbitrary Napoleon’s quasigroup by the formula (11).

2. Equilateral triangles and midpoints

We first state an auxiliary lemma.

Lemma 2.1. In an idempotent medial quasigroup (Q, ·), identity (1) is equivalent with either of the identities

ab · ba = ba · b,  (8)
ab · ca = ba · cb.  (9)

Proof. Using Toyoda’s theorem, the quasigroup can be represented as a · b = a + φ(b − a) in an Abelian group (Q, +) with automorphism φ. The identities (1), (8) and (9) are seen to be equivalent with 3φ^{2} − 3φ + 1_{Q} = 0. □

The following observation will turn out as an algebraic statement of Napoleon’s theorem.

Corollary 2.2. If (Q, ·) is a Napoleon’s quasigroup and a, b, c ∈ Q, then

ab · ca = ac · ba = ba · cb = bc · ab = ca · bc = cb · ac.  (10)

Proof. Follows from (9) by using mediality (3). □

Let (Q, ·) be a Napoleon’s quasigroup. By a triangle we mean an ordered triple of points (a, b, c) ∈ Q^{3}. Using the binary operation we can define equilateral triangles.

Definition 2.3. The triangle (a, b, c) is called left equilateral if ab = bc = ca holds. This is denoted by Δ(a, b, c) or Δ_{o}(a, b, c), where o = ab = bc = ca is the centroid. Similarly, (a, b, c) is called right equilateral if ba = cb = ac = o holds. This is denoted by ∇(a, b, c) or ∇_{o}(a, b, c).

Positive and negative orientation cannot be distinguished in this abstract setting. In the quasigroup (ℂ, ·) defined by a · b = (1 − q)a + qb for q = \frac{1}{2} + \frac{i\sqrt{3}}{6}, left equilateral triangles are positively oriented and right equilateral triangles are negatively oriented, and vice versa for q = \frac{1}{2} − \frac{i\sqrt{3}}{6}. Here are some properties of the ternary relations Δ and ∇.

Proposition 2.4. The statements Δ_{o}(a, b, c), Δ_{o}(b, c, a), Δ_{o}(c, a, b), ∇_{o}(a, c, b), ∇_{o}(c, b, a) and ∇_{o}(b, a, c) are equivalent.

Proof. Obvious from the definition. □

Because of this equivalence, the next two propositions and some other results in the sequel are stated only for left equilateral triangles. Analogous results hold for right equilateral triangles.

Proposition 2.5. If ab = bc = o, then ca = o and Δ_{o}(a, b, c) holds.

Proof. If bc = o, then c = b \ o. We have ca · o = (b \ o)a · o =^{(7)} (ba ·

o) \(oa·o) = (ba·ab) \(oa·o) =^{(8)} (ab·a) \(oa·o) = oa \(oa·o) = o =^{(2)} oo.

By canceling o from the right we get ca = o. □

Proposition 2.6. For all a, b ∈ Q there is a unique c ∈ Q such that Δ(a, b, c) holds.

Proof. Denote o = ab and c = b \ o. Then, ab = bc = o and, according to Proposition 2.5, Δ_{o}(a, b, c) holds. From Δ(a, b, c) we see that c = b \ ab, so c is uniquely determined by a and b. □

Equilateral triangles can also be defined in hexagonal quasigroups [11]. However, in that context centroids of equilateral triangles cannot be expressed explicitly, making them less suitable for proving Napoleon-like theorems. In [8], midpoints were defined in arbitrary medial quasigroups by using parallelograms. Because of [9, Theorem 12], this is equivalent with the following more direct definition in idempotent medial quasigroups.

Definition 2.7. Let (Q, ·) be an idempotent medial quasigroup. The point m ∈ Q is the midpoint of the pair of points (a, b) ∈ Q^{2}, denoted by M(a, m, b), if am · mb = ab holds.

Figure 2. The midpoint relation M(a, m, b).

Here is a characterization, making symmetry of the midpoint relation in a and b apparent.

Proposition 2.8. In an idempotent medial quasigroup (Q, ·), M(a, m, b) is equivalent with ma · bm = m.

Proof. Using Toyoda’s representation theorem, both of the equations am · mb = ab and ma · bm = m are easily seen to be equivalent with a + b = 2m in the underlying Abelian group (Q, +). □

Corollary 2.9. In an idempotent medial quasigroup, M(a, m, b) holds if and only if M(b, m, a) holds.

Proof. Follows from Proposition 2.8 and mediality (3). □

Generally, it is not true that each pair of points has a unique midpoint. To prove this, we need division by 2 in the underlying Abelian group. Our first example of Napoleon’s quasigroup constructed from the complex numbers has this property. However, if an example is defined in the same way from a field of characteristic 2, then every m ∈ Q is the midpoint of the pairs (a, a), while pairs (a, b) with a ≠ b do not possess midpoints. However, if two sides (a, b) and (a, c) of a triangle (a, b, c) in a medial quasigroup possess midpoints, then the third side (b, c) also has a midpoint [8, Theorem 40].

3. Napoleon’s theorem and its relatives

The last few claims of the previous section hold in general idempotent medial quasigroups. Now we turn back to Napoleon’s quasigroups.

Theorem 3.1 (Napoleon’s theorem). Let (a, b, c) be an arbitrary triangle in a Napoleon’s quasigroup (Q, ·). Then, ∇_{o}(ab, bc, ca) and Δ_{o}(ba, cb, ac) hold for some o ∈ Q.

Proof. This is a direct consequence of Definition 2.3 and Corollary 2.2. □

It is known that the centroid of a triangle in the Euclidean plane coincides with the centroid of its Napoleon triangles. This motivates the following definition.

Definition 3.2. The centroid of an arbitrary triangle (a, b, c) ∈ Q^{3} in a Napoleon’s quasigroup (Q, ·) is the point C(a, b, c) = ab · ca.

The point o in Napoleons’s theorem is precisely C(a, b, c). Corollary 2.2 implies that C(a, b, c) = C(d, e, f) for any permutation (d, e, f) of (a, b, c). Of course, if Δ_{o}(a, b, c) or ∇_{o}(a, b, c), then C(a, b, c) = o. Furthermore, Proposition 2.8 can now be reinterpreted as

M(a, m, b) ⟺ C(m, a, b) = m.

Proposition 3.3. Let (a, b, c) ∈ Q^{3} be a triangle in a Napoleon’s quasigroup and suppose m ∈ Q is the midpoint of (a, b), i.e. M(a, m, b) holds. Then, C(a, b, c) = C(m, m, c).

Proof. According to Proposition 2.8 and Definition 3.2, ma · bm = m,

C(a, b, c) = ab · ca and C(m, m, c) = mm · cm =^{(2)} m · cm. Therefore,

(m·cm)(m·cm) =^{(2)} m·cm =^{(4)} mc·m = mc·(ma·bm) =^{(5)} (mc·ma)(mc·

bm) =^{(5),(10)} (m · ca)(cm · bc) =^{(3)} (m · cm)(ca · bc) =^{(10)} (m · cm)(ab · ca). By

cancelling m·cm from the left we get m·cm = ab·ca, i.e. C(m, m, c) = C(a, b, c). □

In the Napoleon’s quasigroup defined from the Euclidean plane, C(m, m, c) is the point dividing the segment \overline{mc} in the ratio 1 : 2. Therefore, the previous proposition can be interpreted as the wellknown fact that the centroid trisects each median.

In 1868, E. Lemoine [4] posed the problem to construct vertices a, b, c of a triangle, given the vertices a_{1}, b_{1}, c_{1} of the equilateral triangles erected over its sides. A solution was published by L. Kiepert [2] in the following year. Kiepert’s solution is to erect equilateral triangles with vertices a_{2}, b_{2}, c_{2} over the sides of the given triangle and to construct midpoints of (a_{1}, a_{2}), (b_{1}, b_{2}) and (c_{1}, c_{2}). Here is a more precise statement in the setting of Napoleon’s quasigroups.

Theorem 3.4. Let (a, b, c) ∈ Q^{3} be a triangle in a Napoleon’s quasigroup and denote by a_{1}, b_{1}, c_{1} ∈ Q the unique points such that Δ(a_{1}, b, c), Δ(a, b_{1}, c) and Δ(a, b, c_{1}) hold. Furthermore, let a_{2}, b_{2}, c_{2} ∈ Q be the unique points such that Δ(a_{2}, b_{1}, c_{1}), Δ(a_{1}, b_{2}, c_{1}) and Δ(a_{1}, b_{1}, c_{2}) hold. Then, M(a_{1}, a, a_{2}), M(b_{1}, b, b_{2}) and M(c_{1}, c, c_{2}).

To make the proof of this and the next theorem shorter, we introduce a new binary operation ∗ in a Napoleon’s quasigroup (Q, ·). Given a, b ∈ Q, denote by a ∗ b the unique point c such that Δ(a, b, c) holds (see Proposition 2.6). Thus,

a ∗ b = b \ ab = (b \ a)b.  (11)

This new operation is obviously idempotent. Because it is defined by a formula involving multiplication and left division, it is mutually medial with the old operation [12]:

(a ∗ b) · (c ∗ d) = ac ∗ bd.  (12)

As a consequence, the new operation is medial itself and is left and right distributive over the old operation:

a ∗ bc = (a ∗ b)(a ∗ c),  (13)
ab ∗ c = (a ∗ c)(b ∗ c),  (14)
a(b ∗ c) = ab ∗ ac,  (15)
(a ∗ b)c = ac ∗ bc.  (16)

The following properties of the new operation will also be useful.

Lemma 3.5. Let (Q, ·) be a Napoleon’s quasigroup and define a ∗ b = b \ ab. Then, for any a, b, c ∈ Q,

(a ∗ b)a = b(a ∗ b) = ab  (17)

and

ab ∗ ca = bc.  (18)

Proof. From the definition of ∗ we have Δ(a, b, a∗b). By Definition 2.3, this means ab = b(a ∗ b) = (a ∗ b)a. By Napoleon’s theorem and Proposition 2.4, we have Δ(ab, ca, bc) and hence bc = ab ∗ ca. □

As a consequence of identity (17), (Q, ∗) is in fact a hexagonal quasigroup. Dividing ab = (a ∗ b)a by a from the left we get b =

a \(a ∗ b)a =^{(11)} (a ∗ b) ∗ a. This is the defining identity for hexagonal quasigroups (see [10]). Now we can also prove Theorem 3.4.

Proof of Theorem 3.4. By the definition of a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2}, we have

a_{1} = b ∗ c, b_{1} = c ∗ a, c_{1} = a ∗ b  (19)

and

a_{2} = b_{1} ∗ c_{1}, b_{2} = c_{1} ∗ a_{1}, c_{2} = a_{1} ∗ b_{1}.  (20)

Multiplying equations (19) by b, c, a respectively and using (17), we get

a_{1}b = bc, b_{1}c = ca, c_{1}a = ab.  (21)

Now we prove M(a_{1}, a, a_{2}) by using Proposition 2.8: aa_{1} · a_{2}a =^{(20)} aa_{1} ·

(b_{1} ∗ c_{1})a =^{(16)} aa_{1} · (b_{1}a ∗ c_{1}a) =^{(21)} aa_{1} · (b_{1}a ∗ ab) =^{(15)} (aa_{1} · b_{1}a) ∗ (aa_{1} ·

ab) =^{(3),(5)} (ab_{1} ·a_{1}a)∗(a·a_{1}b) =^{(19),(21)} (a(c∗a)·a_{1}a)∗(a·bc) =^{(17)} (ca·a_{1}a)∗

(a · bc) =^{(6)} (ca_{1} · a) ∗ (a · bc) =^{(19)} (c(b ∗ c) · a) ∗ (a · bc) =^{(17)} (bc · a) ∗ (a ·

bc) =^{(18)} aa =^{(2)} a. The relations M(b_{1}, b, b_{2}) and M(c_{1}, c, c_{2}) are proved

similarly. □

Branko Grünbaum [1] discovered another Napoleon-like theorem involving midpoints. We state and prove it here in the context of Napoleon’s quasigroups:

Theorem 3.6. Let (a, b, c) ∈ Q^{3} be a triangle in a Napoleon’s quasigroup and denote by a′, b′, c′ ∈ Q the unique points such that ∇(a′, b, c), ∇(a, b′, c) and ∇(a, b, c′) hold. Suppose there are points a_{1}, b_{1}, c_{1} ∈ Q such that Δ(a, b_{1}, c_{1}), Δ(a_{1}, b, c_{1}) and Δ(a_{1}, b_{1}, c) hold; denote the centroids of these three left equilateral triangles by a_{2}, b_{2} and c_{2}. Then, M(b′, a_{1}, c′), M(a′, b_{1}, c′), M(a′, c_{1}, b′) and ∇(a_{2}, b_{2}, c_{2}) hold.

Proof. By the assumptions of the theorem and Proposition 2.4, we have

a′ = c ∗ b, b′ = a ∗ c, c′ = b ∗ a  (22)

and

a = b_{1} ∗ c_{1}, b = c_{1} ∗ a_{1}, c = a_{1} ∗ b_{1}.  (23)

The relation M(b′, a_{1}, c′) follows from Proposition 2.8: a_{1}c′·b′a_{1} =^{(22)} a_{1}(b∗

a)·(a∗c)a_{1} =^{(15),(16)} (a_{1}b∗a_{1}a)·(aa_{1}∗ca_{1}) =^{(12)} (a_{1}b·aa_{1})∗(a_{1}a·ca_{1}) =^{(23)} (a_{1}(c_{1}∗

a_{1}) · aa_{1}) ∗ (a_{1}a · (a_{1} ∗ b_{1})a_{1}) =^{(17)} (c_{1}a_{1} · aa_{1}) ∗ (a_{1}a · a_{1}b_{1}) =^{(5),(6)} (c_{1}a ·

a_{1}) ∗ (a_{1} · ab_{1}) =^{(23)} (c_{1}(b_{1} ∗ c_{1}) · a_{1}) ∗ (a_{1} · (b_{1} ∗ c_{1})b_{1}) =^{(17)} (b_{1}c_{1} · a_{1}) ∗ (a_{1} ·

b_{1}c_{1}) =^{(18)} a_{1}a_{1} =^{(2)} a_{1}. Proofs of the relations M(a′, b_{1}, c′) and M(a′, c_{1}, b′)

are similar.

By Definition 2.3, the centroids of Δ(a, b_{1}, c_{1}), Δ(a_{1}, b, c_{1}) and Δ(a_{1}, b_{1}, c) are a_{2} = b_{1}c_{1}, b_{2} = c_{1}a_{1} and c_{2} = a_{1}b_{1}. According to Propositions 2.4 and 2.5, to prove ∇(a_{2}, b_{2}, c_{2}) it suffices to show b_{2}a_{2} = c_{2}b_{2}, i.e. c_{1}a_{1} · b_{1}c_{1} = a_{1}b_{1} · c_{1}a_{1}. This follows directly from Corollary 2.2. □

The preceding theorem is actually a kind of converse of Grünbaum’s original theorem [1]. Grünbaum assumed a_{1}, b_{1}, c_{1} to be the midpoints of (b′, c′), (a′, c′), (a′, b′) and proved that (a, b_{1}, c_{1}), (a_{1}, b, c_{1}) and (a_{1}, b_{1}, c) are equilateral triangles. This is not true in general Napoleon’s quasigroups. It may happen that a′, b′, c′ coincide and then any a_{1}, b_{1}, c_{1} ∈ Q would be midpoints in a Napoleon’s quasigroup constructed from a field of characteristic 2. However, given a and b_{1}, there is only one c_{1} such that Δ(a, b_{1}, c_{1}) holds.

Floor van Lamoen [3] proved a generalization of Napoleon’s theorem. Here is a slightly modified version in our setting. Napoleon’s theorem is the special case (a_{1}, b_{1}, c_{1}) = (c_{2}, a_{2}, b_{2}).

Theorem 3.7. Let Δ(a_{1}, a_{2}, a_{3}), Δ(b_{1}, b_{2}, b_{3}) and Δ(c_{1}, c_{2}, c_{3}) be equilateral triangles in a Napoleon’s quasigroup (Q, ·). Denote by z_{i} = C(a_{i}, b_{i}, c_{i}), i = 1, 2, 3, and d_{1} = C(a_{1}, b_{2}, c_{3}), e_{1} = C(a_{2}, b_{3}, c_{1}), f_{1} = C(a_{3}, b_{1}, c_{2}), d_{2} = C(a_{1}, b_{3}, c_{2}), e_{2} = C(a_{2}, b_{1}, c_{3}), f_{2} = C(a_{3}, b_{2}, c_{1}). Then, Δ_{o}(z_{1}, z_{2}, z_{3}), Δ_{o}(d_{1}, e_{1}, f_{1}) and Δ_{o}(d_{2}, e_{2}, f_{2}) hold for some o ∈ Q.

Proof. Since the three triangles are left equilateral, a_{1}a_{2} = a_{2}a_{3} = a_{3}a_{1}, b_{1}b_{2} = b_{2}b_{3} = b_{3}b_{1} and c_{1}c_{2} = c_{2}c_{3} = c_{3}c_{1}. Denote o = (a_{1}a_{2} ·b_{1}b_{2})(c_{1}c_{2} · a_{1}a_{2}). Because of Proposition 2.5, it suffices to show that o = z_{1}z_{2} = z_{2}z_{3} = d_{1}e_{1} = e_{1}f_{1} = d_{2}e_{2} = e_{2}f_{2}. This follows by repeated application of mediality, e.g.

\begin{matrix}z_{1}z_{2} = & (a_{1}b_{1} · c_{1}a_{1})(a_{2}b_{2} · c_{2}a_{2}) =^{(3)} (a_{1}b_{1} · a_{2}b_{2})(c_{1}a_{1} · c_{2}a_{2}) =^{(3)} \\ = & (a_{1}a_{2} · b_{1}b_{2})(c_{1}c_{2} · a_{1}a_{2}) = o, \\ d_{1}e_{1} = & (a_{1}b_{2} · c_{3}a_{1})(a_{2}b_{3} · c_{1}a_{2}) =^{(3)} (a_{1}b_{2} · a_{2}b_{3})(c_{3}a_{1} · c_{1}a_{2}) =^{(3)} \\ = & (a_{1}a_{2} · b_{2}b_{3})(c_{3}c_{1} · a_{1}a_{2}) = (a_{1}a_{2} · b_{1}b_{2})(c_{1}c_{2} · a_{1}a_{2}) = o, \\ d_{2}e_{2} = & (a_{1}b_{3} · c_{2}a_{1})(a_{2}b_{1} · c_{3}a_{2}) =^{(3)} (a_{1}b_{3} · a_{2}b_{1})(c_{2}a_{1} · c_{3}a_{2}) =^{(3)} \\ = & (a_{1}a_{2} · b_{3}b_{1})(c_{2}c_{3} · a_{1}a_{2}) = (a_{1}a_{2} · b_{1}b_{2})(c_{1}c_{2} · a_{1}a_{2}) = o. \end{matrix}

The proofs of z_{2}z_{3} = e_{1}f_{1} = e_{2}f_{2} = o are analogous. □

The duals of Theorems 3.4, 3.6 and 3.7, obtained by exchanging Δ with ∇, are also true.

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Department of Mathematics, University of Zagreb, Bijenička 30, HR-10000 Zagreb, Croatia

E-mail address: krcko@math.hr