The Wayback Machine - https://web.archive.org/web/20091027100800/http://www.geocities.com/timeparadox/ismi_azam.html

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The� conjecture of� Ismi Azam

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����������� Attention !� This is NOT a theorem. This is a conjecture.�We have tried and forced this conjecture for months, it had worked with�ZERO percent of error. That is, the area of the regular polygon is exactly same of� the function "A",� up to last five decimal digits which is the official degree of correctness for the software of Cinderella. However we could't be able to force it more than this much of digital correctness, because it is the last limit for this trial software.�

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Case; regular Triangle

a_triangle

Click on this link and download the related winzip� archive that comes with it. You may trust that there is NO any malicious stuff in it. While operating the program.

After unzipping the executable program, play and move the four green points to see that stated facts are a reality. To see things in action click the small arrow at lower

left and make the point P orbiting.

n = 3 ,������ Number of sides of regular polygon

1-� "C" is the center of the circumcircle of regular polygon.

�2-� "P" is any point on the circumcircle.

�3-� From all the vertices of regular polygon, rays are drawn to "P".

�4-� There is a function defined as "A" given on this page. It is related to "S".

�5-� "A" is invariant as "P" moves on circumcircle and values are equal to area of Polygon.

�6-� The locus of the midpoints of all rays pass from an invariant small circle.

�7-� Small circle passes from the main center "C". Area of small circle is equal to (1/4) of the area of the circumcircle.

�8-� When��� n�→ ∞�� ,��� A → π R2��� (area of circumcircle)

�9-� Radius of circumcircle = R

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�Drawn rays are�� (d , e , f)

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"U"� is called "THE CENTRAL BINOM�AL COEFFICIENT"

Provided that� !!!����� 1 ≤ k ≤ (n-1)���

Under these circumstances, the function "A" is equal to the area of the regular polygon with� %100 accuracy.�

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Case; square

b_square

Click on this link and download the related winzip� archive that comes with it. You may trust that there is NO any malicious stuff in it. While operating the program.

After unzipping the executable program, play and move the four green points to see that stated facts are a reality. To see things in action click the small arrow at lower

left and make the point P orbiting.

n = 4 ,������ Number of sides of regular polygon

�1-� "C" is the center of the circumcircle of regular polygon.

�2-� "P" is any point on the circumcircle .

�3-� From all the vertices of regular polygon, rays are drawn to "P".

�4-� There is a function defined as "A" given on this page. It is related to "S".

�5-� "A" is invariant as "P" moves on circumcircle and values are equal to area of Polygon.

�6-� The locus of the midpoints of all rays pass from an invariant small circle.

�7-� Small circle passes from the main center "C". Area of small circle is equal to (1/4) of the area of the circumcircle.

�8-� When��� n�→ ∞�� ,��� A → π R2��� (area of circumcircle)

�9-� Radius of circumcircle = R

�Drawn rays are�� (a , b , c , d)

�

"U"� is called "THE CENTRAL BINOM�AL COEFFICIENT"

Provided that� !!!����� 1 ≤ k ≤ (n-1)�

Under these circumstances, the function "A" is equal to the area of the regular polygon with� %100 accuracy.�

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Case; regular pentagon

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c_pentagon

Click on this link and download the related winzip� archive that comes with it. You may trust that there is NO any malicious stuff in it. While operating the program.

After unzipping the executable program, play and move the four green points to see that stated facts are a reality. To see things in action click the small arrow at lower

left and make the point P orbiting.

n = 5 ,������ Number of sides of regular polygon

�1-� "C" is the center of the circumcircle of regular polygon.

�2-� "P" is any point on the circumcircle .

�3-� From all the vertices of regular polygon, rays are drawn to "P".

�4-� There is a function defined as "A" given on this page. It is related to "S".

�5-� "A" is invariant as "P" moves on circumcircle and values are equal to area of Polygon.

�6-� The locus of the midpoints of all rays pass from an invariant small circle.

�7-� Small circle passes from the main center "C". Area of small circle is equal to (1/4) of the area of the circumcircle.

�8-� When��� n�→ ∞�� ,��� A → π R2��� (area of circumcircle)

�9-� Radius of circumcircle = R

�Drawn rays are�� (a , b , c , d , e)

�

"U"� is called "THE CENTRAL BINOM�AL COEFFICIENT"

Provided that� !!!����� 1 ≤ k ≤ (n-1)�

Under these circumstances, the function "A" is equal to the area of the regular polygon with� %100 accuracy.�

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Case; regular hexagon

d_hexagon

Click on this link and download the related winzip� archive that comes with it. You may trust that there is NO any malicious stuff in it. While operating the program.

After unzipping the executable program, play and move the four green points to see that stated facts are a reality. To see things in action click the small arrow at lower

left and make the point P orbiting.

n = 6 ,������ Number of sides of regular polygon

�1-� "C" is the center of the circumcircle of regular polygon.

�2-� "P" is any point on the circumcircle .

�3-� From all the vertices of regular polygon, rays are drawn to "P".

�4-� There is a function defined as "A" given on this page. It is related to "S".

�5-� "A" is invariant as "P" moves on circumcircle and values are equal to area of Polygon.

�6-� The locus of the midpoints of all rays pass from an invariant small circle.

�7-� Small circle passes from the main center "C". Area of small circle is equal to (1/4) of the area of the circumcircle.

�8-� When��� n�→ ∞�� ,��� A → π R2��� (area of circumcircle)

�9-� Radius of circumcircle = R

�Drawn rays are�� (a , b , c , d , e , f)

�

"U"� is called "THE CENTRAL BINOM�AL COEFFICIENT"

Provided that� !!!����� 1 ≤ k ≤ (n-1)����

Under these circumstances, the function "A" is equal to the area of the regular polygon with� %100 accuracy.�

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Case; regular heptagon

e_heptagon

Click on this link and download the related winzip� archive that comes with it. You may trust that there is NO any malicious stuff in it. While operating the program.

After unzipping the executable program, play and move the four green points to see that stated facts are a reality. To see things in action click the small arrow at lower

left and make the point P orbiting.

n = 7 ,������ Number of sides of regular polygon

�1-� "C" is the center of the circumcircle of regular polygon.

�2-� "P" is any point on the circumcircle .

�3-� From all the vertices of regular polygon, rays are drawn to "P".

�4-� There is a function defined as "A" given on this page. It is related to "S".

�5-� "A" is invariant as "P" moves on circumcircle and values are equal to area of Polygon.

�6-� The locus of the midpoints of all rays pass from an invariant small circle.

�7-� Small circle passes from the main center "C". Area of small circle is equal to (1/4) of the area of the circumcircle.

�8-� When��� n�→ ∞�� ,��� A → π R2��� (area of circumcircle)

�9-� Radius of circumcircle = R

�Drawn rays are�� (a , b , c , d , e , f , g)

�

"U"� is called "THE CENTRAL BINOM�AL COEFFICIENT"

Provided that� !!!����� 1 ≤ k ≤ (n-1)����

Under these circumstances, the function "A" is equal to the area of the regular polygon with� %100 accuracy.�

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Case; regular octagon

f_octagon

Click on this link and download the related winzip� archive that comes with it. You may trust that there is NO any malicious stuff in it. While operating the program.

After unzipping the executable program, play and move the four green points to see that stated facts are a reality. To see things in action click the small arrow at lower

left and make the point P orbiting.

n = 8 ,������ Number of sides of regular polygon

�

�1-� "C" is the center of the circumcircle of regular polygon.

�2-� "P" is any point on the circumcircle .

�3-� From all the vertices of regular polygon, rays are drawn to "P".

�4-� There is a function defined as "A" given on this page. It is related to "S".

�5-� "A" is invariant as "P" moves on circumcircle and values are equal to area of Polygon.

�6-� The locus of the midpoints of all rays pass from an invariant small circle.

�7-� Small circle passes from the main center "C". Area of small circle is equal to (1/4) of the area of the circumcircle.

�8-� When��� n�→ ∞�� ,��� A → π R2��� (area of circumcircle)

�9-� Radius of circumcircle = R

�

�Drawn rays are�� (a , b , c , d , e , f , g , h)

�

"U"� is called "THE CENTRAL BINOM�AL COEFFICIENT"

Provided that� !!!����� 1 ≤ k ≤ (n-1)����

Under these circumstances, the function "A" is equal to the area of the regular polygon with� %100 accuracy.�

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Case; regular Enneagon

g_enneagon

Click on this link and download the related winzip� archive that comes with it. You may trust that there is NO any malicious stuff in it. While operating the program.

After unzipping the executable program, play and move the four green points to see that stated facts are a reality. To see things in action click the small arrow at lower

left and make the point P orbiting.

n = 9 ,������ Number of sides of regular polygon

�1-� "C" is the center of the circumcircle of regular polygon.

�2-� "P" is any point on the circumcircle .

�3-� From all the vertices of regular polygon, rays are drawn to "P".

�4-� There is a function defined as "A" given on this page. It is related to "S".

�5-� "A" is invariant as "P" moves on circumcircle and values are equal to area of Polygon.

�6-� The locus of the midpoints of all rays pass from an invariant small circle.

�7-� Small circle passes from the main center "C". Area of small circle is equal to (1/4) of the area of the circumcircle.

�8-� When��� n�→ ∞�� ,��� A → π R2��� (area of circumcircle)

�9-� Radius of circumcircle = R

�Drawn rays are�� (a , b , c , d , e , f , g , h , j)

������

"U"� is called "THE CENTRAL BINOM�AL COEFFICIENT"�����

Provided that� !!!����� 1 ≤ k ≤ (n-1)�

Under these circumstances, the function "A" is equal to the area of the regular polygon with� %100 accuracy.�

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�How to download source files and Cinderella software ;

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Click on the following link to download my source� (.cdy)� files in "Winzip format".

source files�� But these files require Cinderella software to operate on. To download

the required (free) software click the following link.

Download Cinderella

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GOOD� LUCKS ! for your own studies on this subject....

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