bisect segment

I.10

To bisect a given finite straight line.

—Euclid

../../../../_images/elem.1.prop.10.png

Let AB be the given finite straight line.

Thus it is required to bisect the finite straight line AB.

Let the equilateral triangle ABC be constructed on it, [I.1] and let the angle ACB be bisected by the straight line CD; [I.9]

I say that the straight line AB has been bisected at the point D.

For, since AC is equal to CB, and CD is common,

  • the two sides AC, CD are equal to the two sides BC, CD respectively;

and the angle ACD is equal to the angle BCD;

  • therefore the base AD is equal to the base BD. [I.4]

Therefore the given finite straight line AB has been bisected at D.

## References

[I.1]: /elem.1.3 “Book 1 - Proposition 3” [I.4]: /elem.1.4 “Book 1 - Proposition 4” [I.9]: /elem.1.9 “Book 1 - Proposition 9”