miski

[tex]A_1[/tex] - center ball
[tex]A_1 A_2[/tex] - radius ball
[tex]A_1A_7A_8[/tex] - circular arc
[tex]A_1A_9A_{10}[/tex] - circular arc three times higher than [tex]A_1A_7A_8[/tex] [tex]A_1A_7A_8=A_1A_9A_{11}=A_1A_{11}A_{12}=A_1A_{12}A_{10}[/tex]
[tex]A_1A_2A_5[/tex] - circular arc
[tex]A_1A_2A_3=A_1A_3A_4=A_1A_4A_5[/tex] , points [tex]A_2 , A_3 , A_4 , A_5[/tex] on the best circle the ball (or sphere)
[tex]A_1A_6A_2=A_1A_6A_3=A_1A_6A_4=A_1A_6A_5[/tex] - circular arcs , are circular arcs on a spher

whether the circular arc that looks like straight?

when we look at the top sphere of circular arcs
[tex]A_1A_6A_2,A_1A_6A_3,A_1A_6A_4,A_1A_6A_5[/tex]
- seem straight[tex]A_1A_2,A_1A_3,A_1A_4,A_1A_5[/tex] or [tex]A_6A_2,A_6A_3,A_6A_4,A_6A_5[/tex]

tendon [tex]A_7A_8[/tex] , [tex]A_9A_{10}[/tex] ,[tex]A_{11}A_{12}[/tex] arcs are parallel to each other

- point [tex]A_1[/tex]
- compass , from point [tex]A_1[/tex] , circular arc [tex]A_2A_5[/tex]
- straightedge , in points [tex]A_1 , A_2[/tex] , straight line [tex]A_1A_2[/tex]
- straightedge , in points [tex]A_1 , A_5[/tex] , straight line [tex]A_1A_5[/tex]
- point [tex]A_7[/tex] , requirement [tex]A_1 A_7<\frac{A_1A_2}{3}[/tex]
- compass [tex]A_1 , A_7[/tex] , from point [tex]A_1 [/tex] , point [tex]A_8[/tex]
- straightedge , in points [tex]A_7, A_8[/tex] , straight line [tex]A_7A_8[/tex]
- bisection circular arc [tex]A_2A_5[/tex] , point [tex]B_1[/tex]
- straightedge , in points [tex]A_1, B_1[/tex] , straight line [tex]A_1B_1[/tex] , point [tex]B_2[/tex]

- compass [tex]A_1A_2[/tex] , from point [tex]A_1[/tex] , circular arc [tex]A_9B_3[/tex]
- compass [tex]A_7A_8[/tex] , from point [tex]A_9[/tex] , point [tex]A_{11}[/tex]
- compass [tex]A_7A_8[/tex] , from point [tex]A_{11}[/tex] , point [tex]A_{12}[/tex]
- compass [tex]A_7A_8[/tex] , from point [tex]A_{12}[/tex] , point [tex]A_{10}[/tex]
- straighedge , in point [tex]A_9 ,A_{10}[/tex] , straigt line [tex]A_9A_{10}[/tex]
- bisection circular arc [tex]A_9A_{10}[/tex] , ppint [tex]B_4[/tex]
- straightedge , in points [tex]A_1, B_4[/tex] , straight line [tex]A_1B_4[/tex] , point [tex]B_5[/tex]

To be continued ...

that's the way, without the knowledge of what is happening in the sphere of

- given the angle [tex]C_1C_2C_3[/tex]
- straightedge and compass , straight line [tex]C_2C_3[/tex] , is divided into two equal parts, point [tex]C_4[/tex]
- straightedge and compass , straight line [tex]C_2C_4[/tex] , is divided into two equal parts, point [tex]C_5[/tex]
- compass [tex]C_2C_5[/tex] , from the point [tex]C_2[/tex], point [tex]C_6[/tex]
- straightedge and compass, angle bisection [tex]C_1C_2C_3[/tex] , point [tex]C_7[/tex]
- straightedge , straight line [tex]C_2C_7[/tex]

- compass [tex]C_2C_3[/tex] , from the point [tex]C_2[/tex] , arc [tex]C_3C_1[/tex]
- compass [tex]C_5C_6[/tex] , from the point [tex]C_3[/tex] , point [tex]D_1[/tex]
- compass [tex]C_5C_6[/tex] , from the point [tex]D_1[/tex] , point [tex]D_2[/tex]
- compass [tex]C_5C_6[/tex] , from the point [tex]D_2[/tex] , point[tex]D_3[/tex]
- straightedge , straight line [tex]C_3D_3[/tex]
- straightedge and compass, angle bisection [tex]C_3D_3[/tex] , point [tex]D_4[/tex]
- straightedge , straight line [tex]C_2D_4[/tex] , point [tex]D_5[/tex]

YOU TRY TO KEEP ... Figure down

- straightedge and compass , perpendicular to the line [tex]a_1[/tex] straight line [tex]C_2C_7[/tex]
- compass [tex]C_3D_5[/tex] , in point [tex]C_2[/tex] , points [tex]E_1 and E_2[/tex]
- straightedge and compass , perpendicular to the line [tex]a_2[/tex] line [tex]a_1[/tex] , point [tex]E_3[/tex]
- straightedge and compass , perpendicular to the line [tex]a_3[/tex] line [tex]a_1[/tex] , point [tex]E_3[/tex]
- straighedge , straight line [tex]E_3E_4[/tex] , point [tex]E_5[/tex]
- straightedge and compass , perpendicular to the line [tex]a_4[/tex] straight line [tex]C_5C_6[/tex] , point [tex]E_6[/tex]
- straightedge and compass , perpendicular to the line [tex]a_5[/tex] straight line [tex]C_5C_6[/tex] , point [tex]E_7[/tex]

YOU TRY TO KEEP ... Figure down

- straightedge and compass , perpendicular [tex] b_1[/tex] straight line [tex]C_2D_5[/tex]
- straightedge and compass , perpendicular [tex]b_2[/tex] on the [tex]b_1[/tex] from point [tex]D_3[/tex] , straight line [tex]D_6D_3[/tex]
- straightedge and compass , perpendicular [tex]b_3[/tex] on the [tex]b_1[/tex] from point [tex]D_2[/tex] , straight line [tex]D_7D_2[/tex]

YOU TRY TO KEEP ... Figure down
[tex]F_1[/tex] is located on the arc [tex]C_3C_1 [/tex], [tex]C_3F_1=C_1F_1[/tex]

- straightedge , straight line [tex]C_2F_1[/tex] , [tex]C_2F_1=C_2C_3[/tex]
- compass [tex]C_2E_5[/tex] , from point [tex]C_2[/tex] , point[tex]F_3[/tex]
- straightedge and compass , straight line the normal to [tex]C_2F_3[/tex]
- compass [tex]D_6D_3[/tex] , from point [tex]C_2[/tex] , point[tex]F_4[/tex]
- straightedge ,straight line extension [tex]C_2F_4[/tex]
- compass [tex]D_7D_2[/tex] , from point [tex]C_2[/tex] , point [tex]F_5[/tex]
- straightedge and compass , normal from point [tex]F_5[/tex] na duž [tex]C_2F_1[/tex] , point [tex]F_6[/tex]

Solution - in the picture below

- compass [tex]C_2F_6[/tex] , from point [tex]E_6[/tex] , point [tex]A_{12}[/tex]
- compass [tex]C_2F_6[/tex] , from point [tex]E_7 [/tex], point [tex]A_{13}[/tex]
- straightedge , semi-line [tex]C_2A_{11}[/tex]
- straightedge , semi-line [tex]C_2A_{12}[/tex]

trisection is complete, any error !!!

this is true for angles [tex]180^o<\alpha<0^o [/tex], larger angles of first division of the [tex]180^o[/tex]

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

valid for the odd [tex]a={3,5,7,9,11,...}[/tex]

Proper ninth angle

- straight line [tex]A_1A_2[/tex]
- straightedge and compass ,[tex]\frac{A_1A_2}{10}[/tex] , point [tex]A_4[/tex] , [tex]a+1[/tex] , [tex]a=9. followed by .9+1=10[/tex]
- straightedge and compass , [tex]A_1A_3[/tex] normal [tex]A_1A_2 [/tex] , angle [tex]C_3C_1C_2=90^o[/tex]
- compass [tex]A_1A_4[/tex] , from point [tex]A_5[/tex]
- straightedge , straight line [tex]A_4A_5[/tex]
- straightedge and compass , bisection arc [tex]A_2A_3[/tex] , point [tex]A_6[/tex]

YOU TRY TO KEEP ... Figure down