Can you solve it? Russia’s Prime Minister sets a geometry puzzle

Russia's Prime Minister Mikhail Mishustin visited a sixth-form maths class at one his country's most science-oriented schools to mark the beginning of the school year.
The class was trying to solve a business problem. He asked, "Why do you guys have to do business projects at school?" This is where fundamental knowledge is required.

Then, he wrote the following problem on a blackboard.

Without any measuring tools, construct a perpendicular starting at the red point of the circle and ending at the diameter.

Illustration by ICM 2022

Also, if you have a circle with a diameter and a point on it, how can you draw a line that intersects the diameter at an angle? (As shown in green.

This question has a beautiful side effect. It doesn't allow you to use any measuring devices. To draw straight lines, you can only use an unmarked ruler.

This is how Mishustin's photograph at the top shows you where to start. He is showing that an angle subtended to by a point on the circle at the two ends is a right angle. It may be helpful to have some basic knowledge about triangles. The altitude of a triangular is, for example, the line that runs from one corner to the opposite side at a right angle. The three altitudes of acute triangles will always intersect.

This question is a beautiful puzzle. I also found it interesting that one of Russia's most powerful politicians used geometry questions as a publicity stunt. It's not often that you see a politician solve a math problem by writing it on a board.

Mishustin, a trained engineer, said to sixth-graders: I think it would be a good idea to learn some basics at your age. You will be able solve any problem, even business, when you are proficient in mathematics, physics, and chemistry.

I'll be back with the solution at 5pm UK. Please do not reveal any details.

Every Monday, I create a new puzzle. I am always looking for new puzzles. Send me an email if you have any suggestions.

I am the author of many puzzle books, including the Language Lovers Puzzle book. If you are interested, I can also speak to schools about mathematics and puzzles (restrictions permitting). Please contact me if your school is interested.

The International Congress of Mathematicians 2022 provided the illustrations. Next year, the ICM 2002 will take place in Saint Petersburg.

International mathematical community has criticised the Russian government for what they see as human rights violations against Azat Miftakhov (mathematician, anarchist), who was sentenced earlier this year to six years in prison for alleged hooliganism. More than 3000 mathematicians from around the globe have signed a petition calling for his release.