Hoagland & Bara are both dunces at math, and could never show you the proof of this theorem — the original math was done by the cartographer Erol Torun. However, I enjoy math and I present for your entertainment and edification a simple proof:
Theorem: If a tetrahedron is inscribed within a sphere, with one vertex at the North pole, the other three vertices will lie on the 19.5° South latitude (and of course the same goes for the South pole/North latitude).
Work. It can be shown that the center of a tetrahedron divides its vertical height in the ratio 3:1 — if h is the height, the center lies 3h/4 from each vertex.
The center of a tetrahedron must obviously also be the center of an exscribed sphere, so 3h/4 then becomes the radius of the sphere:
By inspection, sin θ = h/4 × 4/3h = 1/3
∴ θ = arc sin 0.3333 = 19.47277°
Hoagland & Bara are welcome to use this proof in their future work, preferably with attribution to The Emoluments of Mars, although we know they think nothing of "borrowing" other people's work and slapping their own © on it. They can probably draw better diagrams than I can. Cheers.
NOTE THAT this merely confirms the geometric fact. It in no way endorses Hoagland & Bara's mystical ideas about energy appearing at the 19.5° latitudes. That's still as much balderdash as it ever was.
Update 10th April:
Last night, in the course of a two-hour stint on Coast to Coast AM, Hoagland spoke about the "lights" on Ceres, photographed by the Dawn spacecraft back in February.
I rolled my eyes as he said—perhaps inevitably—that the lights are at 19.5°. The text on the C2C web site is as follows:
Enterprise Mission analysis, using the latest refined "Ceres cartography," has discovered that the lights, remarkably, light at ~19.5 degrees North ... and are aligned north of due east by ~19.5 degrees.I can't confirm that so I'm calling it a barefaced lie for the moment. If it later turns out to be true I'll let you know and do a mea culpa.
Research by binaryspellbook shows that the lights are at "about 19 degrees North." OK, good enough. MEA CULPA.