Lexell's theorem

The theorem is named for Anders Johan Lexell, who presented a paper about it c. 1777 (published 1784) including both a trigonometric proof and a geometric one.

[6] There are several ways to prove Lexell's theorem, each illuminating a different aspect of the relationships involved.

The main idea in Lexell's c. 1777 geometric proof – also adopted by Eugène Catalan (1843), Robert Allardice (1883), Jacques Hadamard (1901), Antoine Gob (1922), and Hiroshi Maehara (1999) – is to split the triangle

Jakob Steiner (1827) wrote a proof in similar style to Lexell's, also using Girard's theorem, but demonstrating the angle invariants in the triangle

Euler in 1778 proved Lexell's theorem analogously to Euclid's proof of Elements I.35 and I.37, as did Victor-Amédée Lebesgue independently in 1855, using spherical parallelograms – spherical quadrilaterals with congruent opposite sides, which have parallel small circles passing through opposite pairs of adjacent vertices and are in many ways analogous to Euclidean parallelograms.

There is one complication compared to Euclid's proof, however: The four sides of a spherical parallelogram are the great-circle arcs through the vertices rather than the parallel small circles.

[10] A lemma analogous to Elements I.35: two spherical parallelograms on the same base and between the same parallels have equal area.

(As in Elements, the case where the parallelograms do not intersect on the sides is omitted, but can be proven by a similar argument.)

Another proof using the midpoint circle which is more visually apparent in a single picture is due to Carl Friedrich Gauss (1841), who constructs the Saccheri quadrilateral (a quadrilateral with two adjacent right angles and two other equal angles) formed between the side of the triangle and its perpendicular projection onto the midpoint circle

In 1854 Joseph-Émile Barbier – and independently László Fejes Tóth (1953) – used the polar triangle in his proof of Lexell's theorem, which is essentially dual to the proof by isosceles triangles above, noting that under polar duality the Lexell circle

However, if we allow choice of arc for each pair, then 8 distinct generalized spherical triangles can be made, some with self intersections, of which four might be considered to have the same base

on a sphere, Lexell's theorem holds that the signed surface area of any generalized triangle

bisect each-other and the figure has 2-fold rotational symmetry about the intersection point (so the diagonals each split the parallelogram into two congruent spherical triangles,

Joseph-Émile Barbier later wrote a geometrical proof (1864) which he used to prove Lexell's theorem, by duality; see § Perimeter of the polar triangle above.

Schubert solved this problem by a calculus-based trigonometric approach to show that the triangle of minimal area has its apex at the nearest intersection of the altitude circle and the perpendicular bisector of the base, and the triangle of maximal area has its apex at the far intersection.

In 2019 Vincent Alberge and Elena Frenkel solved the analogous problem in the hyperbolic plane.

[24] In the Euclidean plane, a median of a triangle is the line segment connecting a vertex to the midpoint of the opposite side.

On the sphere, a median of a triangle can also be defined as the great-circle arc connecting a vertex to the midpoint of the opposite side.

However, on the sphere the great-circle arc through one vertex and a point on the opposite side which bisects the triangle's area is, in general, distinct from the corresponding median.

Jakob Steiner used Lexell's theorem to prove that these three area-bisecting arcs (which he called "equalizers") all intersect in a point, one possible alternative analog of the planar centroid in spherical geometry.

[4] Elements I.35 holds that parallelograms with the same base whose top sides are colinear have equal area.

As on the sphere, any generalized circle passing through a pair of antipodal points in hyperbolic geometry is a geodesic.

Chasles, Michel (1837), Aperçu historique sur l'origine et le développment des méthodes en géométrie [Historical overview of the origin and development of methods in geometry] (in French), Brussels: Hayez, Ch.

Maehara, Hiroshi (1999), "Lexell's theorem via an inscribed angle theorem", American Mathematical Monthly, 106 (4): 352–353, doi:10.1080/00029890.1999.12005052 Lexell, Anders Johan (1786), "De proprietatibus circulorum in superficie sphaerica descriptorum" [On the properties of circles described on a spherical surface], Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin), 6: 1782 (1): 58–103, figures tab.

3 Steiner, Jakob (1841), "Sur le maximum et le minimum des figures dans le plan, sur la sphère et dans l'espace général" [On the maximum and the minimum of figures in the plane, on the sphere and in general space], Journal de mathématiques pures et appliquées (in French), 6: 105–170, EuDML 234575 Lebesgue, Victor-Amédée (1855), "Démonstration du théorème de Lexell" [Proof of Lexell's theorem], Nouvelles annales de mathématiques (in French), 14: 24–26, EuDML 96674 Persson, Ulf (2012), "Lexell's Theorem" (PDF), Normat, 60 (3): 133–134 Van Brummelen, Glen (2012), "8.

382–394 Brooks, Jeff; Strantzen, John (2005), "Spherical Triangles of Area π and Isosceles Tetrahedra" (PDF), Mathematics Magazine, 78 (4): 311–314, doi:10.1080/0025570X.2005.11953347, JSTOR 30044179 Sorlin, A. N. J.; Gergonne, Joseph Diez (1825), "Trigonométrie.

Research on spherical trigonometry], Annales de Mathématiques Pures et Appliquées, 15: 273–304, EuDML 80036 Fuss, Nicolas (1788) [written 1784], "Problematum quorundam sphaericorum solutio", Nova Acta Academiae Scientiarum Imperialis Petropolitanae, 2: 70–83 Alberge, Vincent; Frenkel, Elena (2019), "3.

On a problem of Schubert in hyperbolic geometry", in Alberge, Vincent; Papadopoulos, Athanase (eds.

27–46, doi:10.4171/196-1/2 Lei, Kin; Qi, Dongxu; Tian, Xiaolin (2020), "A new coordinate system for constructing spherical grid systems", Applied Sciences, 10 (2): 655, doi:10.3390/app10020655 Shvartsman, Osip Vladimirovich (2007), Комментарий к статье П. В. Бибикова и И. В. Ткаченко «О трисекции и бисекции треугольника на плоскости Лобачевского» [Comment on the article by P. V. Bibikov and I. V. Tkachenko 'On trisection and bisection of a triangle in the Lobachevsky plane'] (PDF), Matematicheskoe Prosveschenie, ser.

3 (in Russian), 11: 127–130 Johnson, Norman W. (1981), "Absolute Polarities and Central Inversion", in Davis, Chandler; Grünbaum, Branko; Sherk, F.A.

Orange triangles △ ABC share a base AB and have the same area. The locus of their variable apex C is a small circle (dashed green) passing through the points antipodal to A and B .
An area formula for spherical triangles analogous to the formula for planar triangles
Lexell's proof by breaking the triangle △ A ∗ B ∗ C into three isosceles triangles
Steiner's proof by constructing a cyclic quadrilateral ◻ A ∗ DB ∗ C inside Lexell's circle
Lemma: Two spherical parallelograms with the same base and between the same parallels have equal area.
Gauss's proof using a Saccheri quadrilateral
Paul Serret's proof by stereographic projection with A projected to the origin
Barbier's proof using the perimeter of a polar triangle △ A′B′C′ , pictured in stereographic projection
In the limit as C → B ∗ along Lexell's circle l the triangle △ ABC degenerates to the lune (shaded orange or purple) with its side tangent to l (here pictured in stereographic projection).
A spherical parallelogram, showing great circles with solid lines and small circles with dashed lines
A foliation of the sphere by Lexell's loci
Steiner's theorem: the "equalizers" (area-bisecting arcs through each vertex) of a spherical triangle all intersect in a point S . (This is distinct from the intersection of the medians G .)
Mapping from spherical area coordinates to barycentric coordinates in the plane
Orange triangles △ ABC share a base AB and area. The locus of their apex C is a line (dashed green) parallel to the base.
Elements I.35
In the half-plane model, antipodal points are reflections into the opposite half-plane (shaded gray). Hyperbolic triangles △ ABC (orange) share a base AB and area. The locus of apex C is a hypercycle (dashed green) passing through points antipodal to A and B .