Re: 2 trains leaving from LA will arive at New York and London at the same time

While it's theoretically possible to build a train which could go that fast, there are several things (friction is most important) which will slow the train down. But problems like these often "idealize" the problem: they make it simpler and ignore all the complicated bits which make real life so difficult. In a problem like this, you should ignore the fact that friction will slow the train down, and imagine a perfect, frictionless, "ideal" train.

You haven't clearly explained the example problem, but I believe I recognize it as a famous physics problem. Imagine we were able to dig a hole straight through the center of the earth and out the other side. (We can't, because the Earth's liquid inside, but that's another "idealization".) Now we drop an elevator down the hole. Gravity will cause it to fall toward the center of the earth, so it will speed up until it passes through the center: at that point, it will start slowing down because it's going away from the center. Eventually (if we ignore friction) it will reach the surface of the other side of the planet. How long does it take to do this?

The answer is 42 minutes. Unfortunately, I can't think of any way to prove this to you without using calculus and trigonometry, two branches of math which you don't know yet. Interestingly, 42 minutes is also the amount of time it takes a low-flying spacecraft (like the Space Shuttle) to go halfway around the Earth.

Even more interestingly, imagine digging a straight tunnel at an angle to the Earth's surface, so that it passes through the earth on a chord rather than a diameter. Imagine replacing the elevator with a train which rolls along this tunnel -- no engines, just rolling downhill. The same thing will happen: it will roll down the slope until it reaches the spot closest to the center, when it will start going uphill (away from the center) and come to rest at the other end of the tunnel. Now, since the tunnel slopes toward the center of the earth gradually, like a ramp -- and since you know that things roll downhill more slowly on gently-tilted ramps -- you would guess that the train would move more slowly than the elevator. And it does. But the length of the chord is always less than the length of the diameter, so the train's tunnel is shorter than the elevator. The train's slower speed exactly cancels out the tunnel's shorter length, and it can be shown that the train takes the same amount of time, 42 minutes, to reach the other side, no matter how you cut the tunnel through the earth.

Unfortunately, I don't know how to prove that to you either, without using trigonometry or calculus. Your question suggests that the proof involves the use of your theorem about intersecting chords... but I proved it without that theorem. There can be several ways to prove something: I suspect that a proof of the "train problem" using your theorem would be very clever indeed... too clever for me.

I realize that's disappointing, but the whole reason these math subjects exist is to let you do problems which are impossible using other sorts of math.

The good news is, you're only two years away from being able to learn this. You should be able to understand this problem with freshman-level college physics... or maybe even your senior year in high school, if you take AP physics and calculus.