Pre RMO Training: Plane geometry with combinatorics

Question 1:

There are 4 possible ways to place three distinct lines in a plane. Two of these configurations involve parallel lines, the other two do not. Draw all these possibilities including the one which encloses a region.

Question 2:

Prove that the sum of any two sides of a triangle is greater than the third side. Hint: Use the following permissible clever argument: the shortest distance joining any two distinct points is given by a straight line joining those two points.

Question 3:

There are 8 possible ways to place 4 distinct lines in a plane. Five of these configurations involve parallel lines; the other three do not. Draw all the possibilities.

Remark: Questions like 1 and 2 are at the heart of combinatorics questions in plane geometry in pre RMO and RMO.

Cheers,
Nalin Pithwa

PS: Prove the parallelogram law: |a+b| \leq |a|+|b|

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