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CTK Exchange
Bui Quang Tuan
Member since Jun-23-07
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Aug-18-08, 08:35 AM (EST) |
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"New Dissection Of Square Into Six Parts ?"
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Dear All My Friends,
I suggest here one Pythagorean theorem proof by dissection of square into six parts. Suppose ABC is right triangle with right angle at C and BC<=AC. On the hypotenuse AB we construct a square ABB'A' on the same side with C with respect to AB. We dissect this square into six parts as following:
- M, N are midpoints of A'B', BB' respectively.
- Because BC<=AC so line BC cuts side segment A'B' at a point, say D and line AC cuts side segment BB' at a point, say E.
- B'' = orthogonal projection of B' on BD
- D' = reflection of D in M
- E' = reflection of E in N
- D'' = orthogonal projection of D' on B'B''
We have six parts of square ABB'A':
1=ABC, 2=BCE, 3=ACDA', 4=CEB'B'', 5=DD'D''B'', 6=B'D'D''
(Please see attach picture file for detail).
Now we can use some rotations to move these six part to get two squares constructed on AC, BC:
1' = rotation of 1 around A by 90
2 and 3 are still in places
4' = rotation of 4 around N by 180
5' = rotation of 5 around B' by 90
6' = rotation of 6 around M by 180
May be it is one new Pythagorean proof?
Thank you and best regards,
Bui Quang Tuan
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Bui Quang Tuan
Member since Jun-23-07
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Aug-19-08, 11:17 PM (EST) |
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1. "RE: New Dissection Of Square Into Six Parts ?"
In response to message #0
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Dear All My Friend,
By this dissection we can show one interesting fact:
In square ABCD two lines L1, L2 are passing through vertices A, B respectively and L1 perpendicular with L2. Two these lines divide square ABCD into 4 parts. The result: there are two parts have same area value.
It is interesting that this fact is true for any regular polygon:
In regular n-gon (n>=3) ABC… two lines L1, L2 are passing through vertices A, B respectively. Moreover angle(L1, L2) = 360/n (center angle of regular n-gon). Resul: L1, L2 divide regular n-gon into 4 parts from which two parts have same area value. (If P is intersection of L1, L2 then one part is triangle APB, other part is a part contains opposite angle APB).
Is there simple proof for this fact?
Thank you and best regards,
Bui Quang Tuan |
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