bisect angle

I.9

To bisect a given rectilineal angle.

—Euclid

../../../../_images/elem.1.prop.9.png

Let the angle BAC be the given rectilineal angle.

Thus it is required to bisect it.

Let a point D be taken at random on AB; let AE be cut off from AC equal to AD; [I.3] let DE be joined, and on DE let the equilateral triangle DEF be constructed; let AF be joined.

I say that the angle BAC has been bisected by the straight line AF.

For, since AD is equal to AE, and AF is common,

  • the two sides DA, AF are equal to the two sides EA, AF respectively.

And the base DF is equal to the base EF;

  • therefore the angle DAF is equal to the angle EAF. [I.8]

Therefore the given rectilineal angle BAC has been bisected by the straight line AF.

references

[I.3]: /elem.1.3 “Book 1 - Proposition 3” [I.8]: /elem.1.8 “Book 1 - Proposition 8”