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Then, BC coinciding with EF,

BA, AC will also coincide with ED, DF;

for, if the base BC coincides with the base EF, and the sides BA, AC do not coincide with ED, DF but fall beside them as EG, GF,
then, given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there will have been constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it. But they cannot be so constructed. [I. 7]

Therefore it is not possible that, if the base BC be applied to the base EF, the sides BA, AC should not coincide with ED, DF;

they will therefore coincide,

so that the angle BAC will also coincide with the angle EDF, and will be equal to it.

If therefore etc.

Q. E. D.

1 2

Proposition 9.

To bisect a given rectilineal angle.

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.

Proposition 10.

To bisect a given finite straight line.

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.

Q. E. F.

1 The text has here “BA, CA.”

2 fall beside them. The Greek has the future, παραλλάξουσι. παραλλάττω means “to pass by without touching,” “to miss” or “to deviate.”

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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