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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,248

Hi;

When the pope asked the great artist Giotto di Bondone to prove his worth as an artist, he drew a perfect circle freehand.

This problem appears in another thread, go there to see more traditional methods of solution.

http://www.mathisfunforum.com/viewtopic.php?id=14214

Here is the problem.

The diagram shows a triangle ABC

with ABC = 80 and ACB = 80

D lies on AC so that DBC = 60

and E lies on AB so that ECB = 50.

To find (by Euclidean geometry) x = EDB

We of course will use geogebra to do it for us. We start by constructing the diagram according to the instructions above.

1) Draw points B and C at (0,0) and (4,0).

2) Draw a line segment between them.

3) Hide the x and y axes.

4) Click the angle with a given size tool and then click C and B. Enter 80° and click okay.

5) Use the tool again only this time click B and then C and set it for clockwise. Press okay.

6) You will notice points B' and C' were created above the line.

7) Draw a line through B and C' and another line through C and B' using the line through two points tool.

8) Scale down the drawing so you can see the triangle that has been formed.

9) Hide B' and C' and use the intersection tool to find the point of intersection of the two newly created lines. It will be marked point A.

10) Hide the two lines BA and CA and the line segment BC so you are just left with the points A,B and C.

11) Use the polygon tool and click points A,B and C. A nice neat triangle will appear.

12) Use the angle with given size tool and click C and then B and enter 60° with counterclockwise. A point C2' and an angle will be created.

13) Use the angle with given size tool again an click B and then C and enter 50° with clockwise. A point B2' and an angle will be created.

14) Draw a line through B and C2' and another line through C and B2'.

15)Hide the angles and the points B2' and C2'

16 Get the point of intersection of one of the new lines with side AC of the triangle. Point D will be created.

17) Do the same thing with the line and AB. Point E will be created.

18) Hide the two lines. Now you should have just a triangle with points D and E.

I am getting tired but there is only a little more.

19) Draw line segment BD and CE.

We want angle EDB shown in red in the first figure.

20) Use the angle tool to measure angle EDB by clicking E, D and then B. Up pops 30° and that is the answer.

I wonder what shape mathematics which had its roots in the geometry of the ancient greeks would have taken if we all could have drawn perfect circles like Giotto? if we all could have freehand like geogebra? It is possible an axiomatic system for mathematics would be much less used or even nonexistent, replaced by experimental mathematics.

More to come.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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