Where is the eighth vertex?

Question

A piece of cheese ABCDEFGH was cut with 6 planar
cuts in the shape of hexahedron, and photographic
picture of this piece of cheese was taken.

http://alexandersemenov.tripod.com/ya/cheese/index.htm

Seven vertixes are visible on this picture, and the last
vetrex H is invisible.

Using straight edge reconstruct position of the last
invisible vertex H on the picrure.

Answer 1

Obviously this problem is easier if a compass could be used, but using a straightedge only is possible. First, it is possible, given a line, to draw another line parallel to it using a straightedge only, although not necessarily through any arbitrary point. I'll leave out the tedious details on how to do this exactly, but it's done with Harmonic Ratios. Now, the method for finding the 8th vertex is based on "taking parallel slices" of the cheese starting at vertex B. The axis of the slicing is arbitrary, but one procedure is as follows:

1) Extend lines BA, BC, BF
2) Draw triangle ACF
3) Draw 2 more triangles slightly larger than triangle ACF, but with sides parallel to it, and with vertices on lines BA, BC, BF
4) Draw 3 pairs of lines through the intersections of the 2 triangles with lines AD, BD, DG, GF, FE, EA. These 3 pairs of lines will form 2 more triangles, again with sides parallel.
5) Draw lines passing through the 3 pairs of vertices of the 2 triangles formed. The 3 lines will intersect at vertex 8.

I give this problem a star because I hadn't realized before that it takes all 7 vertices on the plane to uniquely define the 8th.

Answer 2

Drop a vertical line from point D paralell to AE and CG
Draw a line from E towards the letter H on the diagram, which should be parallel to sides AD and FG
Draw a line from G towards the letter H on the diagram, which should be roughly parallel to CD and EF

There should be one point of intersection of these 3 lines - that will be the vertex H.

Answer 3

hey bro, My vertex is in Pisces ( at 28 degrees 50 ) in 6th house ....but my astrology knowledge hasnt passed beyond the planets yet. I am sorry but I am sure others will give you good answers :)