2

Part of 1987 Flanders Math Olympiad

Problems(1)

2 XY <= AB+ A'B" , midpoints inequality by 3 pairs of parallel lines

Source: Flanders Math Olympiad 1987 p2

Two parallel lines

a

b

meet two other lines

c

d

A

A'

be the points of intersection of

a

c

d

, respectively. Let

B

B'

be the points of intersection of

b

c

d

, respectively. If

X

is the midpoint of the line segment

A A'

Y

is the midpoint of the segment

BB'

|XY| \le \frac{|AB|+|A'B'|}{2}.

geometrygeometric inequality