[Math] How to make a perpendicular construction in 3 moves

euclidean-geometry

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.

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  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)
  7. Craete the segment AD.

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.

Best Answer

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.

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