Prove that lines are colinear
Source: Azerbaijan 2022 Junior National Olympiad
May 14, 2022
geometryJuniorAzerbaijancolinear
Problem Statement
Let be an acute triangle and be the intersection of the meadians of triangle . Let be the foot of the altitude drawn from to . Draw a parallel line such that it is parallel to and one of the points of it is .Donate the point as the intersection of the parallel line and circumcircle . Prove that are co-linear
[asy]
size(6cm);
defaultpen(fontsize(10pt));
pair A = dir(50), S = dir(130), B = dir(200), C = dir(-20), G = (A+B+C)/3, D = foot(A, B, C);draw(A--B--C--cycle, black+linewidth(1));
draw(A--S^^A--D, magenta);
draw(S--D, red+dashed);
draw(circumcircle(A, B, C), heavymagenta);string[] names = {"", "", "","", "",""};
pair[] points = {A, B, C,D,G,S};
pair[] ll = {A, B, C,D, G,S};
int pt = names.length;
for (int i=0; i