** Proposition 8**. *If in an equilateral and equiangular
pentagon straight lines subtend two angles taken in order, they
cut one another in extreme and mean ratio, and their greater segments
are equal to the side of the pentagon.*

For in the equilateral and equiangular pentagon ABCDE let the straight
lines AC, BE, cutting one another at the point H, subtend two angles
taken in order, the angles at A, B; I say that each of them has
been cut in extreme and mean ratio at the point H, and their greater
segments are equal to the side of the pentagon.

For let the circle ABCDE be circumscribed about the pentagon ABCDE.
[IV. 14] [p. 454]

Then, since the two straight lines EA, AB are equal to the two AB,
BC, and they contain equal angles, therefore the base BE is equal
to the base AC, the triangle ABE is equal to the triangle ABC, and
the remaining angles will be equal to the remaining angles respectively,
namely those which the equal sides subtend. [I. 4]

Therefore the angle BAC is equal to the angle ABE; therefore the
angle AHE is double of the angle BAH. [I. 32]

But the angle EAC is also double of the angle BAC, inasmuch as the
circumference EDC is also double of the circumference CB; [III.
28, VI. 33] therefore the angle HAE is equal to the angle AHE; hence
the straight line HE is also equal to EA, that is, to AB. [I. 6]

And, since the straight line BA is equal to AE, the angle ABE is
also equal to the angle AEB. [I. 5]

But the angle ABE was proved equal to the angle BAH; therefore the
angle BEA is also equal to the angle BAH.

And the angle ABE is common to the two triangles ABE and ABH; therefore
the remaining angle BAE is equal to the remaining angle AHB; [I.
32] therefore the triangle ABE is equiangular with the triangle
ABH; therefore, proportionally, as EB is to BA, so is AB to BH.
[VI. 4]

But BA is equal to EH; therefore, as BE is to EH, so is EH to HB.

And BE is greater than EH; therefore EH is also greater than HB.
[V. 14]

Therefore BE has been cut in extreme and mean ratio at H, and the
greater segment HE is equal to the side of the pentagon.

Similarly we can prove that AC has also been cut in extreme and
mean ratio at H, and its greater segment CH is equal to the side
of the pentagon.

Q. E. D. [p. 455]