Find number of existing parallelograms
Source: Iran MO Second Round 1994, P5 and Canada MO 1991, P1
November 26, 2010
geometryparallelogramcombinatorics proposedcombinatorics
Problem Statement
The sides of an equilateral triangle are divided into equal parts For each point on a side, we draw the lines parallel to other sides of the triangle e.g. for we have the following diagram:
[asy]
unitsize(150);
defaultpen(linewidth(0.7));
int n = 3; /* # of vertical lines, including AB */
pair A = (0,0), B = dir(-30), C = dir(30);
draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));
label("",A,W); label("",C,NE); label("",B,SE);
for(int i = 1; i < n; ++i) {
draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);
draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);
draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);
}
[/asy]
For each find the number of existing parallelograms.