# misli

#### msbiljanica

- 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

#### msbiljanica

- 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

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#### msbiljanica

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

#### msbiljanica

- 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

#### HallsofIvy

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

#### msbiljanica

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

#### msbiljanica

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