Geometry for Elementary School/Bisecting an angle

BISECT ANGLE $$\angle ABC $$
 * 1) Use a compass to find points D and E, equidistant from the vertex, point B.
 * 2)  Draw the line $$\overline{DE}$$. [[Image:Geom bisect angle 04.png|396 px]]
 * 3) Construct an equilateral triangle on $$\overline{DE}$$ with third vertex F and get $$\triangle DEF $$. (Lines DF and EF are equal in length). [[Image:Geom bisect angle 05.png|396 px]]
 * 4)   Draw the line $$\overline{BF}$$. [[Image:Geom bisect angle 06.png|396 px]]

Claim

 * 1) The angles $$\angle ABF $$, $$\angle FBC $$ equal to half of $$\angle ABC $$.

The proof

 * 1) $$\overline{DE}$$ is a segment from the center to the circumference of $$\circ B,\overline{BD} $$ and therefore equals its radius.
 * 2) Hence, $$\overline{BE}$$ equals $$\overline{BD}$$.
 * 3) $$\overline{DF}$$ and $$\overline{EF}$$ are sides of the equilateral triangle $$\triangle DEF $$.
 * 4) Hence, $$\overline{DF}$$ equals $$\overline{EF}$$.
 * 5) The segment $$\overline{BF}$$ equals to itself
 * 6) Due to the Side-Side-Side congruence theorem the triangles $$\triangle ABF $$ and  $$\triangle FBC $$ congruent.
 * 7) Hence, the angles $$\angle ABF $$, $$\angle FBC $$ equal to half of $$\angle ABC $$.

Note
We showed a simple method to divide an angle to two. A natural question that rises is how to divide an angle into other numbers. Since Euclid's days, mathematicians looked for a method for trisecting an angle, dividing it into 3. Only after years of trials it was proven that no such method exists since such a construction is impossible, using only ruler and compass.

Exercise

 * 1) Find a construction for dividing an angle to 4.
 * 2) Find a construction for dividing an angle to 8.
 * 3) For which other number you can find such constructions?

Geometria per scuola elementare/Bisettrice di un angolo