# How to make a perpendicular construction in 3 moves?

I've been playing Euclid: The Game for some time now. I'm quite addicited to it, trying to get all the records now. Suprisingly, I'm not able to get a record for some really early level. In Level 4 (http://euclidthegame.com/Level4/), I'm trying to get the record for primitive tools. Primitive tools means that you can only use the first 5 tools (compass and straigthedge), and not any tool that you unlock in later levels.

The current record is 3 moves, but I really don't get how that could be possible. The best I can do is 4 moves.

1. Create a random point on the line B
2. Create a circle with center A and radius AB.
3. Create a point at the intersection of the line and the circle
4. Create a circle with center C and radius BC
5. Create a circle with center B and radius BC
6. Create a point at the intersection of the circles (D)

Point creation is not counted as a move, so this is 4 moves, but how can you do this possibly in 3 moves ?

Oh, one more thing, the goal of this level is: Construct a line (segment) that goes through point A and that is perpendicular to the given line segment.

• It seems that "Angle Bisector" is among the allowed tools. When I use that, the system congratulates me to using minimal number of moves ... Jul 7, 2014 at 9:59
• Just for fun, a solution with 1 move: Put a point $B$ at infinity and draw a cirlce with radius $\bar{AB}$, see here... Jul 7, 2014 at 10:02
• @HagenvonEitzen Yeah, I already have that medal, but I need the medal that you get when you get a congratulation message because you used a minimal number of primitive moves. Jul 7, 2014 at 10:03
• Isn't the angle bisector exactly the construction (in four moves) described above?
– HSN
Jul 7, 2014 at 10:09
• @GerryMyerson 5 tools, 1. construct point on object 2. construct point on intersection 3. construct segment 4. construct a ray (extend a segment) 5. compass Jul 7, 2014 at 10:11

I'm the developer of Euclid: The Game. I'm glad you like the game, don't worry about that you can't get this solution in 3 moves.

Only very recently this record is set. No-one in the top 10 of the highscores has been able to do this, in fact, from all the scores that are submitted, no-one has ever done this in 3 moves.

Some user noted in the comments that it was possible in 3 moves, well I didn't believe it, but he showed me his solution, and it is actually quite simple (but that is true for a lot of things in mathematics).

I'm not going to give the solution here, I'm sure one time someone will spoil it in the comments, and maybe you get an answer here, that is fine with me as well, but I don't want to be responsible for spoiling my own game !

Well, I could give a hint. Your first move shouldn't be to create a random point on the line.

• I almost didn't believe until I finally realized the solution. Now I'm thinking whether to post another hint as an answer or let other people figure out the solution themselves.
– JiK
Jul 7, 2014 at 11:09
• @JiK I'm fine with it if you want to post another hint. Jul 7, 2014 at 11:18
• yeaaaaaaaaah, got it !! Jul 7, 2014 at 11:20
• I'm almost certain that I learned this in elementary or middle school, then totally forgot it until I saw your hint... (though I didn't spend too much time thinking about it) Jul 7, 2014 at 12:06
• It's a pity the choice of the first point can't be entirely random, since it might already lie on the perpendicular. Apart from that, a very nice solution!
– HSN
Jul 7, 2014 at 12:37

Hint:

If you draw three points on a circle in a certain way, you get a right angle.

Name of the theorem helpful for this solution:

From wikipedia:

if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ∠ABC is a right angle

• Welcome to Math.StackExchange! Generally, it is best to put a bit more than just a link in your answer (if the link rots, so does your answer, after all). Can you elaborate a bit on what the link says?
– user88319
Sep 15, 2015 at 5:51

Solution

1. Create a random point $$B$$ outside the line, and make a circle from $$B$$ to $$A$$. By construction, this circle will intersect with the line at two points, where $$A$$ is one of the intersections, and there should be another intersection. 2. Let the other intersection of the circle and the line be $$C$$, make a line from $$C$$ to $$B$$, which extends and intersects the circle at $$D$$. 3. Make the line $$AD$$. It will be the perpendicular line.

• Please clarify "make a circle from B to A". Your phrase "the other intersection" doesn't make sense, because no first intersection was mentioned; please clarify. May 14, 2022 at 6:06
• @RosieF the circle will intersect with the line at two points, the first one is A and the second one is what I referred to as "the other intersection". Jul 28, 2022 at 6:50