Prove that RS || Delta [Iran Second Round 1990]
Source:
December 1, 2010
geometry proposedgeometry
Problem Statement
Let be a parallelogram. The line meets the lines and at and respectively. Let be the intersection point of the lines and let be intersection point of the lines Prove that
[asy]
import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); pen qqzzcc = rgb(0,0.6,0.8); pen wwwwff = rgb(0.4,0.4,1);
draw((2,2)--(6,2),qqzzcc+linewidth(1.6pt)); draw((6,2)--(4,0),qqzzcc+linewidth(1.6pt)); draw((-1.95,(+12-2*-1.95)/2)--(12.24,(+12-2*12.24)/2),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0+3*-1.95)/3)--(12.24,(-0+3*12.24)/3),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0-0*-1.95)/6)--(12.24,(-0-0*12.24)/6),qqzzcc+linewidth(1.6pt)); draw((4,0)--(4,4),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((2,2)--(8.14,0),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((-1.95,(+32.56-4*-1.95)/4.14)--(12.24,(+32.56-4*12.24)/4.14),qqzzcc+linewidth(1.6pt));
dot((0,0),ds); label("", (0,-0.3),NE*lsf); dot((4,0),ds); label("", (4.02,-0.33),NE*lsf); dot((2,2),ds); label("", (1.81,2.07),NE*lsf); dot((6,2),ds); label("", (6.16,2.08),NE*lsf); dot((3,3),ds); label("", (2.97,3.22),NE*lsf); dot((5,1),ds); label("", (4.99,1.19),NE*lsf); label("", (1.7,3.76),NE*lsf); dot((6,0),ds); label("", (5.9,-0.33),NE*lsf); dot((4,2),ds); label("", (4.02,2.08),NE*lsf); dot((4,4),ds); label("", (3.94,4.12),NE*lsf); dot((8.14,0),ds); label("", (8.2,0.09),NE*lsf); clip((-1.95,-6.96)--(-1.95,4.99)--(12.24,4.99)--(12.24,-6.96)--cycle);
[/asy]