Theorem of the gnomon

Gnomon: Theorem of the Gnomon: green area = red area, ${\displaystyle ABFPGD}$

${\displaystyle |AHGD|=|ABFI|,\,|HBFP|=|IPGD|}$

The theorem of the gnomon or theorem of the supplementary parallelograms is a statement about parallelograms and describes the equality of area between two parallelograms of a gnomon .

statement

In a parallelogram with a point on the diagonal , the parallel to through the side in and the side in and the parallel to the side through cuts the side in and the side in The Gnomon theorem then states that the parallelograms and as well as and are equal in area . ${\displaystyle ABCD}$ ${\displaystyle P}$ ${\displaystyle AC}$ ${\displaystyle AD}$ ${\displaystyle P}$ ${\displaystyle CD}$ ${\displaystyle G}$ ${\displaystyle AB}$ ${\displaystyle H}$ ${\displaystyle AB}$ ${\displaystyle P}$ ${\displaystyle AD}$ ${\displaystyle I}$ ${\displaystyle BC}$ ${\displaystyle F}$ ${\displaystyle HBFP}$ ${\displaystyle IPGD}$ ${\displaystyle ABFI}$ ${\displaystyle AHGD}$

The L-shaped figure formed by the two overlapping parallelograms and is called the gnomon. The parallelograms and are called supplementary parallelograms or complements (to the inner parallelograms on the diagonal and ). ${\displaystyle ABFI}$ ${\displaystyle AHGD}$ ${\displaystyle HBFP}$ ${\displaystyle IPGD}$ ${\displaystyle PFCG}$ ${\displaystyle AHPI}$

Derivation

Gnomon's theorem can be proved quite easily if one looks at the areas of the starting parallelogram and the parallelograms on the diagonal. On the one hand, their difference corresponds exactly to the common area of ​​the two supplementary parallelograms and, on the other hand, they are all halved by the diagonal, so the following applies:

${\displaystyle |IPGD|={\frac {|ABCD|}{2}}-{\frac {|AHPI|}{2}}-{\frac {|PFCG|}{2}}=|HBFP|}$

Applications and extensions

Geometrical construction of a division

Transfer of the division ratio of the route AB to the route HG: ${\displaystyle {\tfrac {|AH|}{|HB|}}={\tfrac {|HP|}{|PG|}}}$

The Gnomon theorem can be used to construct a new rectangle or parallelogram of equal area for a given rectangle or parallelogram (in the sense of constructions with compasses and ruler ). This also enables the geometrical construction or representation of the division of two numbers, that is, for two numbers given as route lengths, a new route is constructed whose length corresponds to the quotient of the two numbers (see drawing). Another application is the transfer of a division ratio from one route to another (see drawing).

The lower parallelepiped enclosing the diagonal and its complements , and , which have the same volume: ${\displaystyle \mathbb {A} }$ ${\displaystyle \mathbb {B} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {D} }$ ${\displaystyle |\mathbb {B} |=|\mathbb {C} |=|\mathbb {D} |}$

An analogous statement to the sentence about the gnomon can be formulated in three dimensions for parallelepipeds . Here is a point on the space diagonal of the parallelepiped and one looks at the three planes running through , which are parallel to the outer surfaces of the parallelepiped. Together with the outer surfaces, these form a division of the parallelepiped into eight smaller parallelepipeds. Two of these parallelepipeds enclose the space diagonal and touch each other . These two are now bordered by three of the remaining six parallelepipeds, which here play the role of supplementary parallelograms or complements in the two-dimensional. For each of the two parallelepipedes lying on the diagonal there are three complements and it is now the case that the volumes of these three complements are the same. ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$

General theorem about nested parallelograms

general theorem:
green area = blue area - red area

The Gnomon theorem is a special case of a general statement about nested parallelograms with a common diagonal. For a given parallelogram, consider any inner parallelogram that also has a diagonal. Furthermore, one forms the parallelograms and , whose sides are all parallel to the sides of the outer parallelogram and which have a corner point in common with the inner parallelogram . The difference between the areas of these two parallelograms then corresponds to the area of ​​the inner parallelogram, i.e. the following applies: ${\displaystyle ABCD}$ ${\displaystyle AFCE}$ ${\displaystyle AC}$ ${\displaystyle GFHD}$ ${\displaystyle IBJF}$ ${\displaystyle ABCD}$ ${\displaystyle AFCE}$

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

The theorem of the gnomon is obtained as a borderline case of this statement if one considers the parallelogram , which has degenerated into a straight line and whose corner points are all on the diagonal . Then, in particular, the common corner point of the paralleliograms and lies on the diagonal and their area difference is 0, that is, they have the same area. ${\displaystyle AFCE}$ ${\displaystyle AC}$ ${\displaystyle GFHD}$ ${\displaystyle IBJF}$

Historical

The Gnomon theorem is already described in Euclid's Elements (approx. 300 BC) and there plays an important role in the derivation of various other theorems. The Gnomon theorem is Proposition 43 in the first book of elements. It is formulated there as a statement about parallelograms without using the term gnomon itself. Euclid then introduces this later as the second definition in the second book. Other statements in which the gnomon and its properties play an important role are Proposition 6 in Book II, Proposition 29 in Book VI and Proposition 1, 2, 3 and 4 in Book XIII.

literature

• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: With harmonious proportions to conic sections: pearls of classical geometry . Springer 2016, ISBN 9783662530344 , pp. 190–191
• George W. Evans: Some of Euclid's Algebra . The Mathematics Teacher, Volume 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, Volume 36, No. 1 (Jan. 1929), pp. 32-34 ( JSTOR )
• Paolo Vighi, Igino Aschieri: From Art to Mathematics in the Paintings of Theo van Doesburg . In: Vittorio Capecchi (Hrsg.), Massimo Buscema (Hrsg.), Pierluigi Contucci (Hrsg.), Bruno D'Amore (Hrsg.): Applications of Mathematics in Models, Artificial Neural Networks and Arts . Springer, 2010, ISBN 9789048185818 , pp. 601-610, in particular pp. 303-306

Web links

Commons : Gnomons (geometry)  - collection of images, videos and audio files

• Theorem of the gnomon and definition of the gnomon in Euclid's elements

Individual evidence

1. ↑ ^{a b} Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: With harmonious proportions to conic sections: pearls of classical geometry . Springer 2016, ISBN 9783662530344 , pp. 190–191
2. ^ ^{A b c} William J. Hazard: Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon . The American Mathematical Monthly, Volume 36, No. 1 (Jan. 1929), pp. 32-34 ( JSTOR )
3. ↑ Johannes Tropfke : History of Elementary Mathematics Level Geometry - Volume 4: Level Geometry . Walter de Gruyter, 2011, ISBN 9783111626932 , pp. 134-135
4. ^ ^{A b} Roger Herz-Fischler: A Mathematical History of the Golden Number . Dover, 2013, ISBN 9780486152325 , pp. 35-36
5. ↑ Paolo Vighi, Igino Aschieri: From Art to Mathematics in the Paintings of Theo van Doesburg . In: Vittorio Capecchi (Hrsg.), Massimo Buscema (Hrsg.), Pierluigi Contucci (Hrsg.), Bruno D'Amore (Hrsg.): Applications of Mathematics in Models, Artificial Neural Networks and Arts . Springer, 2010, ISBN 9789048185818 , pp. 601-610, in particular pp. 303-306
6. ↑ George W. Evans: Some of Euclid's Algebra . The Mathematics Teacher, Volume 20, No. 3 (March, 1927), pp. 127-141 ( JSTOR )