BISECT ANGLE ${\displaystyle \angle ABC}$

1. Use a compass to find points D and E, equidistant from the vertex, point B.
2. Draw the line ${\displaystyle {\overline {DE}}}$.
   
   
   
   
3. Construct an equilateral triangle on ${\displaystyle {\overline {DE}}}$ with third vertex F and get ${\displaystyle \triangle DEF}$. (Lines DF and EF are equal in length).
   
   
   
   
4. Draw the line ${\displaystyle {\overline {BF}}}$.
   
   
   
   

Claim

1. The angles ${\displaystyle \angle ABF}$, ${\displaystyle \angle FBC}$ equal to half of ${\displaystyle \angle ABC}$.

The proof

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

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?