Earlier today I set you the following puzzle, which was a challenge Russia’s Prime Minister, Mikhail Mishustin, gave to a class of Russian sixth formers earlier this month.
Construct a perpendicular from the (red) point on the circle to the diameter, without using any measuring devices.
In other words, given a circle with a diameter marked on it, and a point on the circle, can you find a way to draw a line from the point that hits the diameter at a right angle. (As marked in green above.)
What makes this question interesting is the banning of measuring devices. Compass and marked ruler are verboten. All you are allowed is an unmarked ruler to draw straight lines.
I also gave you the following two pieces of information: 1) The angle subtended by a point on a circle to the two ends of a diameter is a right angle. 2) The altitude of a triangle is the line from a corner that meets the opposing side at a right angle. In acute triangles the three altitudes will always intersect.
From reading the comments at the bottom of the original article it seems that few people took my hint about the altitudes of a triangle.
STEP 1 Draw lines from the point to both ends of the diameter. Chose another point on the circle and do the same. The angles from both of these points to the ends of the diameter are right angles, as illustrated.
STEP 2 Extend the lines that go through both points until they meet, as illustrated. Consider this point of intersection as the corner of a “big” triangle whose opposite side is the diameter of the circle. Two of the altitudes of this big triangle are already drawn: these are the thin blue lines that go from each of the two ends of the diameter to the opposite sides. We know that altitudes intersect, so when we draw a line from the top of the big triangle though the intersection of the two lines below it, that must also be an altitude, which means that it hits the diameter at a right angle.
STEP 3 Now we have constructed a line perpendicular to the diameter, our final task is to draw a line parallel to it one that goes through the red point. To do this, draw a line from where our perpendicular touches the circle, through the red dot, to an extension of the diameter.
STEP 4 Draw a line from the other intersection of the perpendicular and the circle, and join it to the point on the diameter marked by the previous step. This line must intersect the circle at the mirror point of the red dot. The line that joins them is the perpendicular we were after.
Ta dah! And here is Russian PM Mishustin’s drawing of the solution.
I hope you enjoyed today’s puzzle. I’ll be back with another one in two weeks.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.
Thanks to the International Congress of Mathematicians (ICM) 2022 for use of the illustrations. The ICM 2022 will be held in Saint Petersburg next year.