Maria's Puzzle Book Reloaded [Puzzle 26]

[VyseGolbez]

After doing the last step to solve the puzzle of the Sorcerer of Curiousity, the Voyager gestured for the others to listen to him: “Solving this together with you was very fun, and it reminded me that we never got to the end of this book full of puzzles I have here. In fact, instead of forcing you to compete against each other, I think this form of cooperatively solving puzzles is far more enjoyable. Away with the time limits! Away with the points! Let’s just have a bit of intellectual fun!”

So yeah, this is a continuation of this thread, as I’m changing the rules though I’m making a new thread. As I have far too many puzzles to make a new topic for each one, I’ll instead post a new one here once an old one has been solved. Unlike for picto’s puzzle, number crunching is required for some puzzles, but just like picto’s puzzle, they are all solvable with the knowledge you got in school. Also, I honestly don’t feel like continuing to get rid of the christmas theme these puzzles had originally, so I hope you don’t mind.

You can always ask questions for each puzzle, and it’s sufficient if you just bold your questions, no need to use pink or anything like that (of course, you can use pink if you want). I won’t guarantee to answer every question, though.

These puzzles also come with a difficulty rating, with 1 being the easiest and 6 being the hardest.

Other than that, I’ll use this OP to quickly link to each puzzle, since this topic has the potential to be quite big once some puzzles have been solved. (Puzzles 1 to 20 are all in the old thread)

Links to Puzzles

Puzzles 1 to 5
Puzzles 6 to 10
Puzzles 11 to 15
Puzzles 16 to 20
Puzzle 21
Puzzle 22
Puzzle 23
Puzzle 24
Usagi’s Guest Puzzle
midsummer’s Guest Puzzle
Puzzle 25
Puzzle 26

[VyseGolbez]

Puzzle 21: Circles don’t lie

Difficulty: 3

Priest Ringlet is practicing a Nativity play with some children from his parish. For that purpose he drew some lines on the floor. First a big circle. Then he drew three chords of same length into the circle, and the intersection points of these chords form a triangle. Every chord is 12 meters long and is separated into three segments by the other two chords. The middle segments of each cord form the previously mentioned triangle. The corners of the triangle are the points where the three kings are supposed to stand in the beginning. Caspar and Melchior have a distance of 4 meters, Melchior and Balthasar stand 7 meters apart, and Balthasar and Caspar have a distance of 5 meters between them. The manger with the child inside is positioned in the centre of the circle.

Does this manger stand inside the triangle, outside the triangle, or on one of the sides of the triangle?

[akafa123]

I’m not at home right now but I think it’s definitely not on one of the side because that would make 3 diameters intersect and that would be just a point not triangle
I drew triangle first and extended chords infinitely and then just placed circle on top
I found radius of a circle r = (chord lenght^2/(4height of a triangle) + height of a triangle)/2 = around 6,12 so 12,24 in diameter
I checked using ruler and also drew it on the paper and I guess it’s inside

[VyseGolbez]

r = (chord lenght^2/(4height of a triangle) + height of a triangle)/2

I’m curious how you got to this formula.

[akafa123]

It was one of the theorems in our geometry textbook and I had it in my school notebook, lol I wanted to throw it away.

[VyseGolbez]

Okay, the algebra is correct there, but following your picture, h is not the height of the triangle, but the remaining part of the radius after subtracting the height (since the height of the triangle in that image is labeled with “r-h”). Besides, you don’t know h, so how did you calculate that? What did you insert for h?

[akafa123]

Omg I’m stupid. I inserted the height of the triangle but it still kinda fits lol

[VyseGolbez]

See I want it exactly fitting, otherwise you might be wrong, you don’t know. In fact, check the lengths of the chords in your construction in the first picture, do they have the same length?

[akafa123]

Yes

I did it with a more precise tool that allows to place in cm, so it does fit

[VyseGolbez]

So for your second construction, measuring with a ruler myself, the chord labeled 4 is 6.7 cm long. The chord labeled 5 is 8 cm long. The chord labeled 7 is 8.5 cm long. Therefore, your construction there can’t be right.

To everyone else reading this, this means this puzzle isn’t solved yet, so feel free to chime in if you have any ideas yourself.

[akafa123]

It’s probably because it was zoomed in but whatever maybe we have to actually explain it mathematically but thats boring

[VyseGolbez]

It has nothing to do with any zooming, your second construction is definitely wrong.

[akafa123]

Is first one too?

[King_Titanite_XV]

man i will have no part in these circle theorems and stuff, i want out

[VyseGolbez]

Yes, the first one is wrong as well.

[akafa123]

Are the angles of triangle
101,5
44,4
34,1

[VyseGolbez]

Yes.

[akafa123]

Well atleast I got that right.
I just tried to make it again and only 1 side didnt fit a little

[akafa123]

Ok I did height of the triangle but I’m not sure if I can calculate it that way and found the sides with sinuses and pythagorean formulas. Is that correct or I’m heading the wrong way again
I guess to draw circle really precise we have to find the length of a line before and after sides of our triangle

[VyseGolbez]

Well the thing is I myself don’t know all the lengths and stuff, as this puzzle can be solved without calculating anything. It might be possible to figure out all the lengths through calculation and then construct the entire thing, which is why I didn’t shut down this approach. But perhaps this red gives others some new ideas, who knows.