1. misli

- 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

2. Re: misli

- 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

3. Re: misli

- 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 $\displaystyle C_3C_1$, $\displaystyle C_3F_1=C_1F_1$

4. Re: misli

- 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

5. Re: misli

Do you have a question? What is supposed to be true of these? What was your point in posting them?

6. Re: misli

- 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

------
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 $\displaystyle A_7A_8=A_9A_{11}=A_{11}A_{12}=A_{12}A_{10}=\frac{A _9A_{10}}{3}$

7. Re: misli

valid for the odd $\displaystyle a={3,5,7,9,11,...}$

Proper ninth angle

- 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