Let’s see:

Most civilizations are in the North so I use Temperate zone North:
23.5 to 66.5 degrees latitude

My calculator has a random button.

Latitude: 66.5 - 23.5 = 43
Longitude: +180 –(-180) = 360

Ran(43) = 23.908
23.5 + 23.908 = 47.408 degrees latitude

Ran(360) = 120.96
120.96 – 180 = -59.04 degrees longitude = 59.04 degrees West longitude

Let’s start between 0 degrees and 26 degrees:

Ran(26) = 19.682
Ran(360) = 138.24 => 138.24 – 180 = -41.76 degrees = 41.76 degrees West longitude

I have drawn the great circle. It crosses over Canada, Alaska, New Guinea, Australia, Antarctica. It is not near any ancient sites. Now since I have been looking into physical landmarks – mountains I looked into the offset error in regards to the tallest mountains in each region. Here it is:

From McKinley Alaska: 300 kilometers
From Kosciuszko Australia: 1565 kilometers
From Puncak Jaya Indonesia: 780 kilometers
From Vinson Massif Antarctica: 3370 kilometers

No match whatsoever, the error is very large. Not surprising.