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S_{ij} +S_{n-1-i,m-1-j} = \frac{2S}{mn}, equal segments

Source: Romania IMO TST 1992 p8

February 19, 2020

combinatorial geometrycombinatorics

Problem Statement

Let

m,n \ge 2

be integers. The sides

A_{00}A_{0m}

and

A_{nm}A_{n0}

of a convex quadrilateral

A_{00}A_{0m}A_{nm}A_{n0}

are divided into

m

equal segments by points

A_{0j}

and

A_{nj}

respectively (

j = 1,...,m-1

). The other two sides are divided into

n

equal segments by points

A_{i0}

and

A_{im}

(

i = 1,...,n -1

). Denote by

A_{ij}

the intersection of lines

A_{0j}A{nj}

and

A_{i0}A_{im}

, by

S_{ij}

the area of quadrilateral

A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}

and by

S

the area of the big quadrilateral. Show that

S_{ij} +S_{n-1-i,m-1-j} = \frac{2S}{mn}

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