Preparing for the proof of Pythagoras' theorem by Euclid
Euclid's Elements. Book 1
Preparing Geogebra
a) In the menu Öptions" or "Vaihtoehdot" choose "Kieli" and then Ënglish(UK)".
b) In the menu Öptions" - "Labelling", check "New points only".
c) In the menu "View" uncheck Äxes", verify that Älgebra window" is checked.
Exercise 1.
Choose two points A and B. Use the tool "regular polygon" to draw squares : ABCD, ADEF, AFGH, AHIJ where J=B and BDKL where K=H and H=L.
The sketch you have got can be thought of as an illustration of AC = Ö2 AB.
Clear the plane.
Exercise 2.
a) Choose three points A, B and C.
b) Draw the line a through A and B.
c) Draw the line b parallel to a going through C.
d) Choose two points D and E on b and one point F on a.
e) Using the tools "parallel line" and ïntersecting point" construct the point G such that DEGF is a parallelogram.
f) Use the tool "Polygon" to draw the surface of the parallelogram DEGF and get its area (in some unknown unit) in the algebra window.
g) Choose an other point H on the line a and construct the parallelogram DEIH. Check that DEGF and DEIH have the same area.
h) Let the point F move on a and check that the area of DEGF does not change.
i) Put the points F, G, H and I on the line a in the order FGHI. Call J the intersection of DH and EG. Note the areas of the parallelograms DEGF and DEIH and of the triangles DHF, GEI, GJH and DEJ. Check that:
 area DEGF
=
 area DHF -  area GJH +  area DEJ
=
 area EIG -  area GJH +  area DEJ
=
 area DEIH
Exercise 3.
a) Construct a triangle ABC with a right angle at A.
b) Construct three squares outside the triangle ABC. Let these squares be BADE, ACFG and CBHI.
c) Draw the segments EC and AH. Why are the triangles BEC and BAH equal ?
d) Draw the parallelogram EBCJ. Why is the area of the triangle BEC half the area of the square BADE?
e) Draw through A the parallel to BH and let it intersect HI in the point K.
f) Construct the parallelogram ABHL. Why is the area of ABHL equal to that of the square BADE?
g) Let M be the intersection point of AK and BC. Why is the area of BHKM equal to that of the square BADE?
h) Show Pythagoras' theorem and use colors to illustrate the proof.


File translated from TEX by TTH, version 3.33.
On 04 Mar 2008, 01:04.