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I'll think about a tangential solution now.

Added 22 minutes later
A tangential solution might be correct, but it needs to be carefully considered...

Added 2 hours and 38 minutes later

Most likely, the trajectory should be such that the tangent to it at any given time passes through both heroes in this problem (the girl will naturally be the point of tangency).

Can you provide evidence that this is the case?
You're probably right, although I had a slightly different solution in mind, and I'm not 100% sure that it will be exactly as you describe. Okay, go ahead and guess. But honestly, I'd also like to see proof of what you wrote. I'll try to check if it's true or not myself.

In short, I don't know how it is with tangents, but if the girl swims north, gradually shifting to the left (assuming the maniac is on the southern side of the lake and started moving counterclockwise), describing an Archimedean spiral so that by the time she lands, she ends up in the south, then she will travel a distance less than a quarter of a circle (calculated using the formula). Accordingly, the maniac with his 4x speed will not have time to complete a full circle.

Something like that...

According to the formulas in this article, if she starts moving north and ends up south, the length of the Archimedean spiral will be k*(f*sqrt(1+f*f)+ln(f+sqrt(1+f*f)))/2 = 2.07918825013928*k
Here, f is the angle, and it is equal to pi/2.
The radius of the lake is, accordingly, k*f, and the circumference of the lake is k*2*pi*f = 9.86960440108936*k.
Yes, indeed, she has to swim less than 4 times the distance the maniac has to run around the entire circle. Okay, Yashchur, you proved everything correctly. Make a wish.

Actually, I saw such a solution myself. Let the radius of the lake be R, the girl's speed be v, and the maniac's speed be 4v. Then the maniac's angular speed around the lake will be 4v/R = v/(R/4). Therefore, at a distance less than R/4 from the center, the girl's angular speed around the center of the lake will be greater, and she, for example, can simply swim towards the maniac from the center at a distance of almost, but less than R/4, and then swim in a circle, not quite reaching the end, and eventually reach a position where she will be on the same line with him, passing through them and the center of the lake, at a distance of (5/4)*R, and accordingly, the shortest distance to the shore for her to swim will be (3/4)R, and the maniac will have to run pi*R to the place where she lands.
(3/4)*R/v < (pi*R)/(4*v), and therefore she will swim to the shore earlier than he runs.

To prove the validity of his theory of relativity to his non-physicist acquaintances, Einstein needed only two objects. What were they?

Probably an easy question...

It's not at all simple... let's start by using familiar objects as examples – not physics concepts.

Yashur, are you referring to Special Relativity? The idea of Special Relativity is the universality of physical laws in inertial frames of reference. This (and how this universality is achieved) solved problems with Maxwell's equations, for example. Something like that. Two objects... What example can be used to illustrate this principle? An apple and a forehead? A dent in the apple, a bruise on the forehead... Regardless of the frame of reference, it still hurts. It's not trivial at all. If the idea is about the constant speed of light... That doesn't quite fit either. If you mean General Relativity... Oh, there Einstein did some recalculations, refined the orbits of planets, and this confirmed General Relativity very well at the time. But it's unlikely that non-physicist friends would understand anything about the refined orbits.

STO or OTO – it doesn’t matter. Friends who are not physicists don’t really care either.:D

The difference is fundamental. Special relativity (SR) can be briefly explained as "nothing can move faster than the speed of light." And general relativity (GR) is like saying, "Let's take a towel and stretch it. In its normal state, it's flat. If we roll a small ball across the towel, it will roll straight and nothing will happen. But now, let's place a large, heavy ball on the stretched towel. The towel will form a depression from the ball. Now, if we roll the small ball across the towel, it will roll into this depression. This is gravity according to Einstein." But in general... maybe Uranus is right. Two objects, two companions. Why specifically two companions? I think it's not by chance. I believe they are also objects.

Ah, there aren't just two of them... Then I'm not sure. But still.

His companions were twins! And using them as an example, he explained the paradox! Probably that's it. Or maybe it was just about abstract twins. My answer is: twins.

My answer is twins.

Incorrect.

Because the twin paradox wasn't invented by Einstein; it was devised and presented to him as a supposed refutation of his special theory of relativity... Yes, something like that did happen. But for now, I don't have any other explanations.

Zlobny Yashur

Hermit

Zlobny Yashur
After thinking for a bit, I came to the conclusion that it won't be possible to swim perpendicularly to the maniac's movement...
Most likely, the trajectory should be such that the tangent to it at any given time passes through both characters in this problem (the girl will naturally be the point of tangency).;)
An Archimedean spiral or something like that...

Regarding this answer, I can say almost the same thing as I did last time.
The answer cannot be considered satisfactory because it is at least very incomplete and not definitive, and at least some explanations should be given as to what this will lead to.
What does "incomplete" mean? In what sense "not definitive"? I already said what it will lead to:

to an increase in the distance traveled by the maniac

The girl will reach the shore before the maniac reaches the landing point.
Now I'll think about the tangent option.
Added 22 minutes ago
The tangent option might be correct, but it needs to be thought through very carefully...

I'll allow myself to return to the riddle about the girl on the lake and the maniac for a bit longer.
Yashur correctly solved the riddle and proved that if the girl swims in an Archimedean spiral, she will reach the shore before the maniac.
But I was still bothered by one question: was I right to reject his answer as incorrect when Yashur suggested that the girl move along the tangent drawn from the maniac to her??? And now I can confidently say that I was!!! If she swims away from him along the tangent, she will not be caught by him! She won't be caught even if she starts moving at a distance of (3/4)R from the shore and (5/4)R from the maniac - I wrote above how she can achieve this. So this option is even worse than swimming straight to the shore.

At first, I tried to derive the equation of the line along which she would swim away from him along the tangent, but honestly, I couldn't even compose its differential equation. There were some terribly cumbersome expressions. :(
And the question remained open for me. Then I decided to try to simulate this situation: will she be caught by him or not??? And now I can definitely say that if she swims away from him along the tangent, she will definitely not be caught, even if she starts swimming from the initial position at a distance of (5/4)R from him, she still won't be caught, and the curve along which she will swim will not be an Archimedean spiral!!! In general, if anyone is interested, they can look at it themselves and see it for themselves!

While I was writing this post, another thought came to me: let's check what will happen if she swims as Yashur suggested at the very beginning - perpendicular to the maniac's movement. I tried it, and it turned out that no matter where she starts moving, whether from the center of the lake or at a distance of (5/4)R from the maniac, in this case she will simply swim in a circle with a radius equal to a quarter of the pond's radius!!!

In general, the lower the girl's speed, the closer the result will be to the mathematical one, but the longer you will have to wait for it. The most accurate result will be with a girl's speed of 1, but you will have to wait quite a long time. :(

http://www.hermit.besaba.com/e/a/girlandmaniac.zip

The idea of the Archimedean spiral came to me right away, but I mistakenly thought that if you move perpendicularly to the maniac or along a tangent, you would get this very spiral...

But will there be an answer about Einstein?:(

No, I definitely don't have a coherent version about Einstein.