Dynamic applications of the parallelogram law and some generalizations of the Pythagoras’ theorem

Pavel Leischner University of South Bohemialeischne@pf.jcu.cz Faculty of Education České Budějovice, CZ

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

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

2 2 212 2 (2 )

2a c d b

Trapezoid ABCD in wich |BD| = |BC|

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

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)

Jaromír Šimša April 2010

2 2 2 2( ) ( )a c b d e f

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

ac bd ef

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

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

Thank you for your attention.