misli

Jan 2012
80
0
1a.png
- given the angle \(\displaystyle C_1C_2C_3\)

- straightedge and compass , straight line \(\displaystyle C_2C_3\) , is divided into two equal parts, point \(\displaystyle C_4\)
- straightedge and compass , straight line \(\displaystyle C_2C_4\) , is divided into two equal parts, point \(\displaystyle C_5\)
- compass \(\displaystyle C_2C_5\) , from the point \(\displaystyle C_2\), point \(\displaystyle C_6\)
- straightedge and compass, angle bisection \(\displaystyle C_1C_2C_3\) , point \(\displaystyle C_7\)
- straightedge , straight line \(\displaystyle C_2C_7\)


- compass \(\displaystyle C_2C_3\) , from the point \(\displaystyle C_2\) , arc \(\displaystyle C_3C_1\)
- compass \(\displaystyle C_5C_6\) , from the point \(\displaystyle C_3\) , point \(\displaystyle D_1\)
- compass \(\displaystyle C_5C_6\) , from the point \(\displaystyle D_1\) , point \(\displaystyle D_2\)
- compass \(\displaystyle C_5C_6\) , from the point \(\displaystyle D_2\) , point\(\displaystyle D_3\)
- straightedge , straight line \(\displaystyle C_3D_3\)
- straightedge and compass, angle bisection \(\displaystyle C_3D_3\) , point \(\displaystyle D_4\)
- straightedge , straight line \(\displaystyle C_2D_4\) , point \(\displaystyle D_5\)


YOU TRY TO KEEP ... Figure down
1b.png
 
Jan 2012
80
0
- straightedge and compass , perpendicular to the line \(\displaystyle a_1\) straight line \(\displaystyle C_2C_7\)
- compass \(\displaystyle C_3D_5\) , in point \(\displaystyle C_2\) , points \(\displaystyle E_1 and E_2\)
- straightedge and compass , perpendicular to the line \(\displaystyle a_2\) line \(\displaystyle a_1\) , point \(\displaystyle E_3\)
- straightedge and compass , perpendicular to the line \(\displaystyle a_3\) line \(\displaystyle a_1\) , point \(\displaystyle E_3\)
- straighedge , straight line \(\displaystyle E_3E_4\) , point \(\displaystyle E_5\)
- straightedge and compass , perpendicular to the line \(\displaystyle a_4\) straight line \(\displaystyle C_5C_6\) , point \(\displaystyle E_6\)
- straightedge and compass , perpendicular to the line \(\displaystyle a_5\) straight line \(\displaystyle C_5C_6\) , point \(\displaystyle E_7\)

YOU TRY TO KEEP ... Figure down
1cc.png
 
Last edited:
Jan 2012
80
0
- straightedge and compass , perpendicular \(\displaystyle b_1\) straight line \(\displaystyle C_2D_5\)
- straightedge and compass , perpendicular \(\displaystyle b_2\) on the \(\displaystyle b_1\) from point \(\displaystyle D_3\) , straight line \(\displaystyle D_6D_3\)
- straightedge and compass , perpendicular \(\displaystyle b_3\) on the \(\displaystyle b_1\) from point \(\displaystyle D_2\) , straight line \(\displaystyle D_7D_2\)

YOU TRY TO KEEP ... Figure down
\(\displaystyle F_1\) is located on the arc \(\displaystyle C_3C_1 \), \(\displaystyle C_3F_1=C_1F_1\)
1d.png
 
Jan 2012
80
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- straightedge , straight line \(\displaystyle C_2F_1\) , \(\displaystyle C_2F_1=C_2C_3\)
- compass \(\displaystyle C_2E_5\) , from point \(\displaystyle C_2\) , point\(\displaystyle F_3\)
- straightedge and compass , straight line the normal to \(\displaystyle C_2F_3\)
- compass \(\displaystyle D_6D_3\) , from point \(\displaystyle C_2\) , point\(\displaystyle F_4\)
- straightedge ,straight line extension \(\displaystyle C_2F_4\)
- compass \(\displaystyle D_7D_2\) , from point \(\displaystyle C_2\) , point \(\displaystyle F_5\)
- straightedge and compass , normal from point \(\displaystyle F_5\) na duž \(\displaystyle C_2F_1\) , point \(\displaystyle F_6\)

Solution - in the picture below
1e.png
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Do you have a question? What is supposed to be true of these? What was your point in posting them?
 
Jan 2012
80
0
- compass \(\displaystyle C_2F_6\) , from point \(\displaystyle E_6\) , point \(\displaystyle A_{12}\)
- compass \(\displaystyle C_2F_6\) , from point \(\displaystyle E_7 \), point \(\displaystyle A_{13}\)
- straightedge , semi-line \(\displaystyle C_2A_{11}\)
- straightedge , semi-line \(\displaystyle C_2A_{12}\)

trisection is complete, any error !!!

this is true for angles \(\displaystyle 180^o<\alpha<0^o \), larger angles of first division of the \(\displaystyle 180^o\)

are you ready for the process of construction of the regular polygon

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HallsofIvy -question
1.if my trisection possible ?
2.from the picture below, if you can figure out which actions the ball (sphere) applied in the plane ?
arc \(\displaystyle A_7A_8=A_9A_{11}=A_{11}A_{12}=A_{12}A_{10}=\frac{A_9A_{10}}{3}\)
3d sfera.png
y3d sfera.png
 
Jan 2012
80
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valid for the odd \(\displaystyle a={3,5,7,9,11,...}\)


Proper ninth angle
2.png
- straight line \(\displaystyle A_1A_2\)
- straightedge and compass ,\(\displaystyle \frac{A_1A_2}{10}\) , point \(\displaystyle A_4\) , \(\displaystyle a+1\) , \(\displaystyle a=9. followed by .9+1=10\)
- straightedge and compass , \(\displaystyle A_1A_3\) normal \(\displaystyle A_1A_2 \) , angle \(\displaystyle C_3C_1C_2=90^o\)
- compass \(\displaystyle A_1A_4\) , from point \(\displaystyle A_5\)
- straightedge , straight line \(\displaystyle A_4A_5\)
- straightedge and compass , bisection arc \(\displaystyle A_2A_3\) , point \(\displaystyle A_6\)

YOU TRY TO KEEP ... Figure down
22.png