I think I've found some errors in this episode. All related to the triangulation.

An isosceles triangle was given with sides of 5191 miles (they said it's the distance between the ISS and Houston and New York), and the top vertex angle: 18 degrees. The goal was to calculate the base of that triangle (that is, the distance between Houston and New York). Their answer was: 1604 miles. This answer seems to be wrong, and here's why:

Knowing the sides of the isosceles triangle and the angle between them, we can calculate the base easily.

For example, we can use the law of cosines: c = root(a^2 + b^2 - 2*a*b*cos(angle)). Substituting c=base, a=b=side, angle=the angle between the sides, we get: base = root( 2*side^2 - 2*side^2*cos(angle)) = side*root(2)*root(1 - cos(1) = 1624.102616... which is around 20 miles greater than 1604 from the episode.

Or, the easier way, you can divide this isosceles triangle by its height onto two identical right triangles, with hypotenuse = 5191 and angle = 18 degrees / 2 = 9 degrees. The height (unknown) will be one of its sides, and the other one will be half the base of the original triangle. The whole base thus could be calculated as base = 2*sin(9)*5191 = 1624.102616... which is, again, too big. But we have the same outcome again, and this is the right answer. The answer in the episode is incorrect.

Then I wondered: What should they screw up to get 1604? :-J

When you calculate 5191*sin(1, you get 1604.107218..., which is approximately 1604 from the episode. So it seems that some smart ass have simply calculated a sine of the vertex angle and multiplied by the side, hoping to get the base in this way :-P Unfortunately it works only in right triangles! It won't work for isosceles. Could it be that the creators don't know basic trigonometry? :-/ I hope not, because this would lower their esteem severely ;-P

But I suspect what was the original idea of the creators. 18 degrees is a half of 36 degrees -- an angle which appears in the so-called "Golden Triangle", one of the arms of a pentagram. The triangle shown in the episode really looks like 36 degrees rather than 18 degrees. The Golden Triangle is isosceles, and it has side-to-base -- the sine of half the angle 36 degrees (that is, 18 degrees) -- equal exactly Phi, the Golden Number 1.618033989... . So let's check: side/base = Fi, thus base = side/Fi = 5191 / 1.618... = 3208.214436... . Half of this number is equal to 1604.107218..., which according to the authors is the base, while in reality it's half the base of 36-degree isosceles. Maybe the authors reasoned in such a way?: "For 36 degrees isosceles it's 3208, so for 18 degrees isosceles it'll be 1604"? (Obviously wrong, because half of the isosceles is right triangle, not another isosceles.) Well.. Their intentions could be right, but the outcome stinks ;-)

But there's more errors :-/

The base of the triangle is straight. But the distance on the globe is measured on the geodesic line, which is an arc of a great circle passing through both of them, because that's the shortest path in elliptical geometry. It's definitely not straight. I've checked the real distance between Houston and New York in several places (eg. Google Maps) and it oscillates between 1416..1419 miles. So the 1604 from the episode is a whole 200 miles too much! Unfortunately, taking the straight distance (through the inside of the Earth) this distance will be not longer, but even shorter than 1416 miles :-P So where did they get 1604 miles? :-P I don't know.