Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem...
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![Page 1: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/1.jpg)
Dynamic applications of the parallelogram law and some generalizations of the Pythagoras’ theorem
Pavel Leischner University of South [email protected] Faculty of Education České Budějovice, CZ
![Page 2: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/2.jpg)
In the basis of parallelogram law is the sum and difference of vectors. Elementary form of the law says (see denotation in figure)
It became familiar since the year 1935 when J. von Neumann showed that the Banach Space in which the equation (1) holds is the Hilbert space.
2 2 2 2
Parallelogram law
2( ) c d a b
In my presentation I would like show that equation (1) in connection with a dynamic geometry is a good tool for students mathematical discovering and for solving various problems.
2 2 2 22( ) (1)c d a b
![Page 3: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/3.jpg)
2 2 2 2 2: ( )v d a x b x 2 2 2 2 (1)d a b ax
(Euclid’s Elements, prop. II.13) 2 2 2cos 2 cosx b d a b ab
(Law of cosines)2 2 2 2 2: ( )v c a x b x
2 2 2 2 (2)c a b ax
2 2 2 22( ) c d a b
(Euclid’s Elements, prop. II.12)
Students are able to derive the parallelogram law independentlyusing the Pythagora‘s theorem
![Page 4: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/4.jpg)
2 2 21 2 2 (2 )2
a c d b
Trapezoid ABCD in wich |BD| = |BC|
![Page 5: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/5.jpg)
Related triangles are congruent iff they are right-angled. In such situation c = d and the parallelogram law expresses the Pyhagora’s theorem.
Related triangles with congruent sides placed perpendicular to each other
![Page 6: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/6.jpg)
For areas A, B, C, D, E and F prove that
3( )D E F A B C
2( )C F A B
2( )A D B C
2( )B E C A
(Dutch MO, 1992)
![Page 7: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/7.jpg)
Jaromír Šimša April 2010
2 2 2 2( ) ( )a c b d e f
![Page 8: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/8.jpg)
a c AL
b d AK
2 2 2 22( )AL AK e f
2 2 2 2( ) ( ) 2( )a c b d e f
![Page 9: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/9.jpg)
ac bd ef
![Page 10: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/10.jpg)
Euclid’s Elements, prop. VI.31
In right-angled triangles the area of a figure on the side opposite the right angle equals the sum of the areas of similar and similarly described figures on the sides containing the right angle.
1 2P P P
![Page 11: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/11.jpg)
I am interested in work with mathematically gifted students and new approaches to teaching mathematics. So I showed in my contribution several examples which are associated with Pythagoras‘ theorem. I hope it could be useful for teachers and students.
A word at the end
![Page 12: Dynamic applications of the parallelogram law and some generalizations of the Pythagoras theorem Pavel Leischner University of South Bohemia](https://reader035.fdocuments.in/reader035/viewer/2022081513/5a4d1b737f8b9ab0599b6384/html5/thumbnails/12.jpg)
Thank you for your attention.