# Theorem of the gnomon

Last updated

The theorem of the gnomon states that certain parallelograms occurring in a gnomon have areas of equal size.

## Theorem

In a parallelogram ${\displaystyle ABCD}$ with a point ${\displaystyle P}$ on the diagonal ${\displaystyle AC}$, the parallel to ${\displaystyle AD}$ through ${\displaystyle P}$ intersects the side ${\displaystyle CD}$ in ${\displaystyle G}$ and the side ${\displaystyle AB}$ in ${\displaystyle H}$. Similarly the parallel to the side ${\displaystyle AB}$ through ${\displaystyle P}$ intersects the side ${\displaystyle AD}$ in ${\displaystyle I}$ and the side ${\displaystyle BC}$ in ${\displaystyle F}$. Then the theorem of the gnomon states that the parallelograms ${\displaystyle HBFP}$ and ${\displaystyle IPGD}$ have equal areas. [1] [2]

Gnomon is the name for the L-shaped figure consisting of the two overlapping parallelograms ${\displaystyle ABFI}$ and ${\displaystyle AHGD}$. The parallelograms of equal area ${\displaystyle HBFP}$ and ${\displaystyle IPGD}$ are called complements (of the parallelograms on diagonal ${\displaystyle PFCG}$ and ${\displaystyle AHPI}$). [3]

## Proof

The proof of the theorem is straightforward if one considers the areas of the main parallelogram and the two inner parallelograms around its diagonal:

• first, the difference between the main parallelogram and the two inner parallelograms is exactly equal to the combined area of the two complements;
• second, all three of them are bisected by the diagonal. This yields: [4]
${\displaystyle |IPGD|={\frac {|ABCD|}{2}}-{\frac {|AHPI|}{2}}-{\frac {|PFCG|}{2}}=|HBFP|}$

## Applications and extensions

The theorem of the gnomon can be used to construct a new parallelogram or rectangle of equal area to a given parallelogram or rectangle by the means of straightedge and compass constructions. This also allows the representation of a division of two numbers in geometrical terms, an important feature to reformulate geometrical problems in algebraic terms. More precisely, if two numbers are given as lengths of line segments one can construct a third line segment, the length of which matches the quotient of those two numbers (see diagram). Another application is to transfer the ratio of partition of one line segment to another line segment (of different length), thus dividing that other line segment in the same ratio as a given line segment and its partition (see diagram). [1]

A similar statement can be made in three dimensions for parallelepipeds. In this case you have a point ${\displaystyle P}$ on the space diagonal of a parallelepiped, and instead of two parallel lines you have three planes through ${\displaystyle P}$, each parallel to the faces of the parallelepiped. The three planes partition the parallelepiped into eight smaller parallelepipeds; two of those surround the diagonal and meet at ${\displaystyle P}$. Now each of those two parallepipeds around the diagonal has three of the remaining six parallelepipeds attached to it, and those three play the role of the complements and are of equal volume (see diagram). [2]

## General theorem about nested parallelograms

The theorem of gnomon is special case of a more general statement about nested parallelograms with a common diagonal. For a given parallelogram ${\displaystyle ABCD}$ consider an arbitrary inner parallelogram ${\displaystyle AFCE}$ having ${\displaystyle AC}$ as a diagonal as well. Furthermore there are two uniquely determined parallelograms ${\displaystyle GFHD}$ and ${\displaystyle IBJF}$ the sides of which are parallel to the sides of the outer parallelogram and which share the vertex ${\displaystyle F}$ with the inner parallelogram. Now the difference of the areas of those two parallelograms is equal to area of the inner parallelogram, that is: [2]

${\displaystyle |AFCE|=|GFHD|-|IBJF|}$

This statement yields the theorem of the gnomon if one looks at a degenerate inner parallelogram ${\displaystyle AFCE}$ whose vertices are all on the diagonal ${\displaystyle AC}$. This means in particular for the parallelograms ${\displaystyle GFHD}$ and ${\displaystyle IBJF}$, that their common point ${\displaystyle F}$ is on the diagonal and that the difference of their areas is zero, which is exactly what the theorem of the gnomon states.

## Historical aspects

The theorem of the gnomon was described as early as in Euclid's Elements (around 300 BC), and there it plays an important role in the derivation of other theorems. It is given as proposition 43 in Book I of the Elements, where it is phrased as a statement about parallelograms without using the term "gnomon". The latter is introduced by Euclid as the second definition of the second book of Elements. Further theorems for which the gnomon and its properties play an important role are proposition 6 in Book II, proposition 29 in Book VI and propositions 1 to 4 in Book XIII. [5] [4] [6]

## Related Research Articles

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist. Three equivalent definitions of parallelepiped are

A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle, tetragon, and 4-gon. A quadrilateral with vertices , , and is sometimes denoted as .

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as  ABCD.

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezium in English outside North America, but as a trapezoid in American and Canadian English. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below.

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.

In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum, typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Euclid's Elements, and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. The term is also applied to the Pythagorean theorem.

In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line that contains their diameters.

In Euclidean geometry, the British flag theorem says that if a point P is chosen inside rectangle ABCD then the sum of the squares of the Euclidean distances from P to two opposite corners of the rectangle equals the sum to the other two opposite corners. As an equation:

Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was published posthumously in 1731.

In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

The quadratrix or trisectrix of Hippias is a curve, which is created by a uniform motion. It is one of the oldest examples for a kinematic curve, that is a curve created through motion. Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem. Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle.

Anne's theorem, named after the French mathematician Pierre-Leon Anne (1806–1850), is a statement from Euclidean geometry, which describes an equality of certain areas within a convex quadrilateral.

Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria, who discovered it.

Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem. Because of the latter the restatement of the Pythagorean theorem in terms of quadrilaterals is occasionally called the Euler–Pythagoras theorem.

## References

• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie. Springer 2016, ISBN   9783662530344, pp. 190–191 (German)
• George W. Evans: Some of Euclid's Algebra. The Mathematics Teacher, Vol. 20, No. 3 (March 1927), pp. 127–141 (JSTOR)
• William J. Hazard: Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon. The American Mathematical Monthly, Vol. 36, No. 1 (January 1929), pp. 32–34 (JSTOR)
• Paolo Vighi, Igino Aschieri: From Art to Mathematics in the Paintings of Theo van Doesburg. In: Vittorio Capecchi, Massimo Buscema, Pierluigi Contucci, Bruno D'Amore (editors): Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer, 2010, ISBN   9789048185818, pp. 601–610