Euclidean Geometric Construction: Trisecting an arbitrary angle

Question

I am an undergraduate student hence I have much lack of math knowledge. I was playing with the euclidean structures and constructions and then I confused myself. I know an algebraic proof that we cannot trisect an angle because the angle of $ \pi/9 $ is not constructable (algebraically proven), but I cannot find the mistake in this math delirium, this math absurdity. In the below procedure I tried to explain what I did and my question is that could this gives us second thoughts about trisecting angles? We know that it can't be truw but could you indicate my mistakes? Thank you in advance and I really apologize about this.

Let’s assume that we have an arbitrary angle with edge at the point A. I followed the below process:

First step: Circle construction. We construct a circle with center point $A$ (using compass), let’s name it $C_A$, which intersects the first side of the angle at point $B$ and the second side of the angle at the point $C$.

Second step: Finding Midpoint of a line segment. We draw the line segment $BC$ which connects the points $B$ and $C$. Using compass, we draw two circles with radius $BC$ and center $B$ and $C$ and we call the $C_B$ and $C_C$ respectively. The circles intersection of these circles are the points $D$ and $E$ and the line segment $DE$ which connects them is the mediator (line segment) of the line segment $BC$, which intersects it at the point $F$.

Third step: Circle construction. We construct one more circle (using compass) with center the point $F$ and radius $FB$. Let’s call it $C_F$.

Fourth step: Trisecting a semicircle (or constructing a half hexagon). Now we have to construct three circles (using compass) with the same radius $FB$ as follows:

• The first circle is created with center $B$, we name it $C_b$, and it intersects the circle $C_F$ at the point $G$.
• The second circle is created with center $G$, we name it $C_G$, and it intersects the circle $C_F$ at the point $H$.
• The third circle is created with center $H$, we name it $C_G$, and it intersects the circle $C_F$ at the point $C$, because the triangles $BGF$, $GFH$ and $HFC$ are equilateral (we know it by construction). (We just trisected the semicircle.)

Fifth step: Rays constructions We draw the rays $AG$ and $AH$, which $AG$ ray starts at the point $A$ and passes through the point $G$ and starts at the point $A$ and $AH$ ray passes through the point $H$. Do the rays $AG$ and $AH$ trisect the angle between the rays $AB$ and $AC$?