A beautiful and challenging geometry construction

I came across this puzzle in Alex Bellos's column on The Guardian. He has a new book that I wanted to share titled "The Language Lover's Puzzle Book." I read a review copy of the math puzzles chapter and it was very fun to work out the puzzles. Links to the book and other work by Alex Bellos below ⬇⬇⬇ Amazon link* amzn.to/3nIqjwH *As an Amazon Associate I earn from qualifying purchases. *I am very curious how many people will read this comment and check out the book. If this is something people are interested in, I will feature books and resources in future videos. But I need data to know for sure. The link includes code to keep track of how many people click on it and buy the book. This is an experiment, so if you are not interested and think it will ruin the channel, do let me know that too by replying to this comment. Anyway, enough of that...back to solving the world's problems, one video at a time ;) Alex Bellos's column on The Guardian www.theguardian.com/profile/alexbellos Alex Bellos website www.alexbellos.com/

The answer is: it's impossible. You can't construct a line with only a straight edge. You also need a pencil or a pen.

Fun fact: There is an infinite number of primes but a finite number of prime ministers.

The question was impressive, but I'm more impressed that the auto-generated subtitles got Presh's name exactly right at the beginning of the video lol.

I solved it the same way. This problem seems hard at first because you don't even know how to start but since you only have a straightedge there's not a lot of things you can construct based on just three points on a circle. That's why the first construction had to be an arbitrary point somewhere on the circle or on the three lines. Then there's also not a lot of new points and when you connect the new points you get a perpendicular line which is already something, but the problem is that it's not from C but from an arbitrary place. So the last step has to be to fix it and construct some point so that it gives you some symmetry.

Assuming the circle was drawn on paper, just fold the paper along AB and push your pencil point through C to the other side of the paper (D). Join CD.

My idea? Draw it on a piece of paper then fold in half to wherever you want CD to be and use the edge of the paper as a ruler

I am 10 grade student from India and I try so much on problems that you bring but finally I end up with nothing but after watching solution I feel like I love mathematics

Me: Show puzzle: Extremely Hard After Solution: Oh! That’s an easy one.

Challenge from Mishustin goes to an international level :)

Really a very interesting geometric problem. Thanks for sharing and educating.

Russians know how to draw a perfect circle with just a stright edge.

Extend AB. Draw a tangent from point C which intersect extended AB at E. From point E draw a tangent opposite side of C. The tangent touch the circle at point F. Join CF which intersect AB at D. CD is perpendicular to AB.

It may be more convenient to pick G as an arbitrary point outside of the circle, and connect it with A and B. Then to use the obtained intersections with the circle to find the orthocenter. And as already mentioned in the comments, connecting CP instead of QC may be a better choice as well.

The reason why it is possible is that all projective transformations of the disk preserving the given diameter also preserve the family of chords orthogonal to that diameter. Given a circle, one can prove that any construction that is possible with a straightedge and a compass can also be done with just a straightedge, provided the problem is invariant under all projective transformations taking the circle to itself.

Math problem from Russian government: 1 million rubles was allocated on a road fixing. If 0.1 million was spent on road fixing, where did 0.9 million go?

Marvellous! Beautiful solution. Watching the solution to this problem was like witnessing the climax of a plot unfold.

I wouldn't have solved that either, but enjoyed watching.....

You can hear when he’s describing the solution how excited he is!

Me: Uses phone's edge for 90° Mishustin: 10 YEARS OF GULAG LABOR CAMP + HARDWORK