Book 13
16
For the same reason, if we join MV, which belongs to a hexagon, MX is also inferred to belong to a pentagon.
16
But LM also belongs to a pentagon; therefore the triangle LMX is equilateral.
16
Similarly it can be proved that each of the remaining triangles of which MN, NO, OP, PL are the bases, and the point X the vertex, is also equilateral.
16
Therefore an icosahedron has been constructed which is contained by twenty equilateral triangles.
16
It is next required to comprehend it in the given sphere, and to prove that the side of the icosahedron is the irrational straight line called minor.
16
For, since VW belongs to a hexagon, and WZ to a decagon, therefore VZ has been cut in extreme and mean ratio at W, and VW is its greater segment; [XIII. 9] therefore, as ZV is to VW, so is VW to WZ.
16
But VW is equal to VE, and WZ to VX; therefore, as ZV is to VE, so is EV to VX.
16
And the angles ZVE, EVX are right; therefore, if we join the straight line EZ, the angle XEZ will be right because of the similarity of the triangles XEZ, VEZ.
16
For the same reason, since, as ZV is to VW, so is VW to WZ, and ZV is equal to XW, and VW to WQ, therefore, as XW is to WQ, so is QW to WZ.
16
And for this reason again, if we join QX, the angle at Q will be right; [VI. 8] therefore the semicircle described on XZ will also pass through Q. [III. 31]
