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.

• Q. E. F.

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