Clever Geek Handbook

Constructing with a compass and a ruler, only ... without a compass

We are all familiar from the school curriculum with constructions using a compass and a ruler. And what will happen if suddenly the compass is lost? Is it possible to build something else non-trivial with the help of one ruler? I bring to your attention a problem, the solution of which brought me many pleasant hours. This is an asterisk problem, so don't be discouraged if you don't find a solution right away. Although one friend of mine did it in five minutes, I think that this is rather an exception to the rule.



So, there is a segment on the plane that we would like to continue. It would seem that you select two points on a segment, attach a ruler and draw a straight line. But here's the bad luck: just on the way of this straight line there is a greasy spot of ketchup, and we don't want to stain the ruler. You need to continue the original straight line behind the spot, without touching it with a ruler.







Let me remind you that there is no compass, only an ungraded ruler remains, with which you can only draw a straight line through the selected two points, nothing else. This is a real challenge without tricks. Fold the leaves, use the second side of the ruler to lay down parallel lines, and also do not need to be cunning in other similar ways.







Update: Since the problem was cracked like a nut in the comments, I publish two of my solutions. Peeping at the answers without trying to solve the problem yourself is unsportsmanlike :)



Solution # 1
, ABC S. , A'B'C', :







AB A'B' P1, ( ABS). P2 — BC B'C', P3 — AC A'C'.



P1, P2 P3 , ABC A'B'C', — .



, P1 P2 , P3, , . , , . ,







Solution # 2
, . ( ) , . , .







— , — . — 3 , . A B — , , , ( 3) , .



1. , .



2. A , .



3. P, Q, R S, , .



4. QP RS, B, .



5. : , . , , .



:)







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