Area formula of Pappus

The area formula of Pappus , also called the Pappus theorem , is a theorem of Euclidean triangular geometry , which goes back to the late antique mathematician Pappus Alexandrinus and which was introduced by him in Book IV of the Mathematical Collections around the year 320 . The formula deals with an essential generalization of the Pythagorean theorem and applies to any triangles , with parallelograms instead of Pythagorean squares .

formulation

Pappos' theorem for triangles: striped area = checkered area

An arbitrary triangle of the Euclidean plane is given . The triangle side opposite the corner point is chosen as the base side of the triangle . ${\displaystyle ABC}$ ${\displaystyle \mathbb {E} ^{2}}$ ${\displaystyle A}$ ${\displaystyle {\overline {BC}}}$

Over the two other sides of the triangle and , in each case opposite the corner points or , let two arbitrary parallelograms and be located, with the intersection of the two straight lines and ${\displaystyle {\overline {AB}}}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle ABDE}$ ${\displaystyle ACFG}$ ${\displaystyle H={DE}\cap {FG}}$ ${\displaystyle DE}$ ${\displaystyle FG}$

The parallelogram lies above the base , and it is assumed that: ${\displaystyle {\overline {BC}}}$ ${\displaystyle BCML}$

(1) The sides and are parallel to the straight line ${\displaystyle {\overline {BL}}}$ ${\displaystyle {\overline {CM}}}$ ${\displaystyle AH}$

(2) The sides , and are of equal length : ${\displaystyle {\overline {AH}}}$ ${\displaystyle {\overline {BL}}}$ ${\displaystyle {\overline {CM}}}$

${\displaystyle |{\overline {AH}}|=|{\overline {BL}}|=|{\overline {CM}}|}$.

Then:

The area of the parallelogram is equal to the sum of the areas of the two parallelograms and . ${\displaystyle BCML}$ ${\displaystyle ABDE}$ ${\displaystyle ACFG}$

In formulas:

${\displaystyle F_{BCML}=F_{ABDE}+F_{ACFG}}$

To the evidence

Proof by shear and displacement

The evidence can be presented as follows:

The starting point is that by the straight fact a division of the Euclidean plane in two closed half-planes is given. ${\displaystyle AH}$ ${\displaystyle \mathbb {E} ^{2}}$

The intersections of these two half-planes with the parallelogram in turn form two parallelograms and , which divide, the intersection of the straight line being with the side and the intersection of the straight line with the side . ${\displaystyle BCML}$ ${\displaystyle {H^{*}}{A^{*}}LB}$ ${\displaystyle {H^{*}}{A^{*}}MC}$ ${\displaystyle BCML}$ ${\displaystyle {H^{*}}}$ ${\displaystyle AH}$ ${\displaystyle {\overline {BC}}}$ ${\displaystyle {A^{*}}}$ ${\displaystyle AH}$ ${\displaystyle {\overline {LM}}}$

By means of shear and parallel displacement - in the respective half-plane! - you can now see that coextensive with and also coextensive with . ${\displaystyle {H^{*}}{A^{*}}LB}$ ${\displaystyle ABDE}$ ${\displaystyle {H^{*}}{A^{*}}MC}$ ${\displaystyle ACFG}$

This can be understood in three sub-steps (see below), whereby the treatment of the two parallelograms and is completely similar. ${\displaystyle ABDE}$ ${\displaystyle ACFG}$

In this way you get the desired identity :

${\displaystyle F_{BCML}=F_{{H^{*}}{A^{*}}LB}+F_{{H^{*}}{A^{*}}MC}=F_{ABDE}+F_{ACFG}}$.

Representation of the partial steps

Using the parallelogram , the sub-steps can be described as follows: ${\displaystyle ABDE}$

Sub-step 1

Within the straight line and bordered - that is, in between! - concluded strip is the parallelogram in an area equal parallelogram sheared in such a manner that the points of the site remain fixed , while the point in the point , the point in the point , the page in the page and the page in the page transition . ${\displaystyle AB}$ ${\displaystyle DE}$ ${\displaystyle {\mathcal {S}}_{AB-DE}}$ ${\displaystyle ABDE}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle D}$ ${\displaystyle I}$ ${\displaystyle E}$ ${\displaystyle H}$ ${\displaystyle {\overline {BD}}}$ ${\displaystyle {\overline {BI}}}$ ${\displaystyle {\overline {AE}}}$ ${\displaystyle {\overline {AH}}}$

Substep 2

Along the straight and always within the line and edged concluded strip the resulting in partial step 1 parallelogram is so moved that a new Zwischenparallelogramm arises where in and in transition. ${\displaystyle AH}$ ${\displaystyle (={A^{*}H^{*}})}$ ${\displaystyle BI}$ ${\displaystyle (=BL)}$ ${\displaystyle AH}$ ${\displaystyle {\mathcal {S}}_{BI-AH}}$ ${\displaystyle I}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle L}$

Substep 3

Inside , the intermediate parallelogram created in sub-step 2 is sheared into the parallelogram in such a way that all points on the side remain fixed. ${\displaystyle {\mathcal {S}}_{BI-AH}}$ ${\displaystyle {H^{*}}{A^{*}}LB}$ ${\displaystyle {\overline {BL}}}$

Relation to the Pythagorean theorem

The Pythagorean theorem is obtained when it is assumed that, firstly, the triangle at right angles with right angles at , with short sides and as well as hypotenuse , and that, secondly, the parallelograms and squares are. ${\displaystyle ABC}$ ${\displaystyle A}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle {\overline {BC}}}$ ${\displaystyle ABDE}$ ${\displaystyle ACFG}$

As can then be seen, the triangles and are both right-angled and congruent to the starting triangle and the straight line coincides with the vertical straight line through on . The parallelogram is therefore a rectangle and even a square. The area formula in this case coincides with the Pythagorean formula ${\displaystyle AGH}$ ${\displaystyle AHE}$ ${\displaystyle ABC}$ ${\displaystyle AH}$ ${\displaystyle A}$ ${\displaystyle {\overline {BC}}}$ ${\displaystyle BCML}$ ${\displaystyle |{\overline {CM}}|=|{\overline {AH}}|=|{\overline {BC}}|}$

${\displaystyle |{\overline {BC}}|^{2}=|{\overline {AB}}|^{2}+|{\overline {AC}}|^{2}}$

together. Furthermore, it can be seen that the above evidence also provides evidence of the Euclidean cathetus theorem .

Demarcation

The local theorem is not identical to the Great Theorem by Pappus , which, however, also goes back to Pappus Alexandrinus.

literature

• Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics . Mathematical Association of America, 2010, ISBN 978-0-88385-348-1 , pp. 77-78 ( excerpt (Google) )
  • Claudi Alsina, Roger B. Nelsen: Enchanting Evidence: A Journey Through the Elegance of Mathematics . Springer Spectrum, Berlin / Heidelberg 2013, ISBN 978-3-642-34792-4 .
• Anna Maria Fraedrich: The sentence group of the Pythagoras (=  textbooks and monographs on didactics of mathematics . Volume 29 ). BI-Wissenschaftsverlag, Mannheim / Leipzig / Vienna / Zurich 1994, ISBN 3-411-17321-1 .
• Theophil Lambacher , Wilhelm Schweizer (Ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools. Geometry . Edition E, 13th edition. Part 2. Ernst Klett Verlag, Stuttgart 1965.
• Howard Eves: Pappus's Extension of the Pythagorean Theorem . In: The Mathematics Teacher , Vol. 51, No. 7 (November 1958), pp. 544-546, JSTOR 27955752

Web links

Commons : Set of Pappos  - collection of images, videos and audio files

• Works by and about Pappus, Alexandrinus  in the German Digital Library
• Hans-Jürgen Elschenbroich: Web publication. (PDF) MNU

References and comments

1. ^ ^{A b} Claudi Alsina, Roger B. Nelsen: Enchanting Evidence: A Journey Through the Elegance of Mathematics . Springer Spectrum, Berlin / Heidelberg 2013, ISBN 978-3-642-34792-4 , p. 91-92 .
2. ↑ ^{a b} Theophil Lambacher , Wilhelm Schweizer (Ed.): Lambacher-Schweizer . Mathematical teaching material for higher schools. Geometry . Edition E, 13th edition. Part 2. Ernst Klett Verlag, Stuttgart 1965, p. 102 .
3. ↑ ^{a b c} Anna Maria Fraedrich: The sentence group of Pythagoras (=  textbooks and monographs on didactics of mathematics . Volume 29 ). BI-Wissenschaftsverlag, Mannheim / Leipzig / Vienna / Zurich 1994, ISBN 3-411-17321-1 , p. 88-89 .
4. ↑ The formulation About ... means that the respective parallelogram has only one side in common with the triangle, so -; in this sense! ; - is outside the triangle.
5. ↑ The order of the points is immaterial for the representation of the straight lines and segments. So for two points it is always and . ${\displaystyle X,Y}$ ${\displaystyle XY=YX}$ ${\displaystyle {\overline {XY}}={\overline {YX}}}$
6. ↑ A segment is parallel to a given straight line if and only if the straight line on which the segment lies and the given straight line are parallel.
7. ↑ The presentation essentially follows that of Alsina and Nelsen (p. 92). Alsina and Nelsen state that their presentation was again taken from the American mathematician Howard Whitley Eves. The proof approach is similar to that of the shear proof of the Pythagorean theorem . The sketch to be found in Lambacher-Schweizer on p. 102 suggests that this approach to prove the area formula was known earlier.
8. ↑ is therefore in the two corresponding open half planes separated . ${\displaystyle \mathbb {E} ^{2}\setminus AH}$
9. ↑ If one assumes a fixed left-right orientation of the Euclidean plane and denotes the closed half-planes for two different parallel straight lines on the left side with or , those on the right side with or and continues with the above. d. A. , the strip in between has the representation . ${\displaystyle \mathbb {E} ^{2}}$ ${\displaystyle g,h\subset \mathbb {E} ^{2}}$ ${\displaystyle {H_{g}}^{-}}$ ${\displaystyle {H_{h}}^{-}}$ ${\displaystyle {H_{g}}^{+}}$ ${\displaystyle {H_{h}}^{+}}$ ${\displaystyle g\subset {H_{h}}^{-}}$ ${\displaystyle {\mathcal {S}}_{g-h}={H_{g}}^{+}\cap {H_{h}}^{-}}$