Dear Mr. Goodfellow,

Many thanks for your interesting letter about the Giza Pyramids. We seem to agree that these structures were not originally built as tombs; although I am sure that they were used as such during the IVth Dynasty; and I happen to think that the great majority of pyramids - including Sekhemkhet's, by the way - were built for that purpose. The wall-paintings you mention really belonged in the adjoining temples... However, barbarian though I may be, the idea of finding the intersections of circles enclosing the three pyramids of Giza, had never crossed my mind before reading your last letter. I had previously given some thought to the smaller of the two ground circles, but the larger one, passing through the S.E. corners, is so disproportionate with its q km. radius, that I rather doubt whether the architect could have accurately drawn it in the way he would have wanted to, on a scale plan of reasonable dimensions. On the other hand, he may have known that the centre of a circle lies at the inter section of the perpendicular bisectors of the chords, and could conceivably have calculated a solution on that basis.
Now as you say, any three points make a circle - even if they are aligned, as some would have it. But since the three pyramids are separated by axial distances in a logical extension of the square bases, and all the dimensions are simply accounted for in this way - the enclosing circles, and their points of intersection, are in my view unlikely to have been controlling factors in the design. At the same time, point of intersection were produced and may have been of interest to the builders - but I would be suprised if the dimensions were at all significant.
As to the intersection of the circle passing through the three apexes, I really think that you should abandon this idea. I see no reason for neglecting the different levels of the three pyramid-bases, since the builders were well aware of these and would without doubt have taken them into account. However, the apexes will be shifted by so much that I don't think there could be any chance of a common intersection with the ground circles. Also, which of the three bases is to be chosen -as the ground level - or perhaps one should find the mean level of the three, or the level of the ground where the circles meet?
Just to set your brains rattling, I enclose a copy of the only data I have of the different levels, taken from Vyse's "Operations carried on at the Pyramids of Gizeh in 1837", Volume II. You will find that the apex of the Second Pyramid is actually higher than that of the Great Pyramid, while the Third Pyramid is likewise raised 40 feet or so. I find it hard to imagine that the architect would ever have attempted a three dimensional trignometrical solution for the intersection of these circles. The plane of the circle through the apexes is not, of course, vertical, but tilted at some large angle to the horizontal - something that I would definitely prefer not to contemplate.
After a number of years of sharpening my wits under pyramids - as it were - my present intention is to try and preserve them. So although I would be happy to confirm or disprove your theories when these are based on level ground, any particularly mind-boggling mental exercise is not going to take my fancy. Thus I might try working out the numbers but for the two ground circles only. I would, however, be interested to know what you are expecting to find from digging at a spot which, from the photos I have of the area looks like a pretty barren stretch of desert?

Yours Sincerely, John Legon