2013 Greece Team Selection Test
Given is a triangle $ABC$.On the extensions of the side $AB$ we consider points $A_1,B_1$ such that $AB_1=BA_1$ (with $A_1$ lying closer to $B$).On the extensions of the side $BC$ we consider points $B_4,C_4$ such that $CB_4=BC_4$ (with $B_4$ lying closer to $C$).On the extensions of the side $AC$ we consider points $C_1,A_4$ such that $AC_1=CA_4$ (with $C_1$ lying closer to $A$).On the segment $A_1A_4$ we consider points $A_2,A_3$ such that $A_1A_2=A_3A_4=mA_1A_4$ where $0<m<\frac{1}{2}$.Points $B_2,B_3$ and $C_2,C_3$ are defined similarly,on the segments $B_1B_4,C_1C_4$ respectively.If $D\equiv BB_2\cap CC_2 \ , \ E\equiv AA_3\cap CC_2 \ , \ F\equiv AA_3\cap BB_3$, $\ G\equiv BB_3\cap CC_3 \ , \ H\equiv AA_2\cap CC_3$ and $I\equiv AA_2\cap BB_2$,prove that the diagonals $DG,EH,FI$ of the hexagon $DEFGHI$ are concurrent.



Diagram[asy]import graph; size(12cm); 

real labelscalefactor = 0.5; /* changes label-to-point distance */

pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ 

pen dotstyle = black; /* point style */ 

real xmin = -7.984603447540051, xmax = 21.28710511372557, ymin = -6.555010307713199, ymax = 10.006614273002825;  /* image dimensions */

pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); 



draw((1.1583842866003107,4.638449718549554)--(0.,0.)--(7.,0.)--cycle, aqaqaq); 

 /* draw figures */

draw((1.1583842866003107,4.638449718549554)--(0.,0.), uququq); 

draw((0.,0.)--(7.,0.), uququq); 

draw((7.,0.)--(1.1583842866003107,4.638449718549554), uququq); 

draw((1.1583842866003107,4.638449718549554)--(1.623345080409327,6.500264738079558)); 

draw((0.,0.)--(-0.46496079380901606,-1.8618150195300045)); 

draw((-3.0803965232149757,0.)--(0.,0.)); 

draw((7.,0.)--(10.080396523214976,0.)); 

draw((1.1583842866003107,4.638449718549554)--(0.007284204967787214,5.552463941947242)); 

draw((7.,0.)--(8.151100081632526,-0.9140142233976905)); 

draw((-0.46496079380901606,-1.8618150195300045)--(8.151100081632526,-0.9140142233976905)); 

draw((-3.0803965232149757,0.)--(0.007284204967787214,5.552463941947242)); 

draw((10.080396523214976,0.)--(1.623345080409327,6.500264738079558)); 

draw((0.,0.)--(3.7376079411107392,4.8751985535596685)); 

draw((-0.7646359770779035,4.164347956460432)--(7.,0.)); 

draw((1.1583842866003107,4.638449718549554)--(5.997084862772141,-1.150964422430769)); 

draw((0.,0.)--(7.966133662513563,1.6250661845198895)); 

draw((-2.308476341169285,1.3881159854868106)--(7.,0.)); 

draw((1.1583842866003107,4.638449718549554)--(1.6890544250513695,-1.624864820496926)); 

draw((2.0395968109217,2.660375186246903)--(2.9561195753832448,0.6030390855677443), linetype("2 2")); 

draw((3.4388364046369224,1.909931693481981)--(1.4816619768719694,0.8229159040072803), linetype("2 2")); 

draw((1.3969966570225139,1.8221911417546572)--(4.301698851378541,0.8775330211014288), linetype("2 2")); 

 /* dots and labels */

dot((1.1583842866003107,4.638449718549554),linewidth(3.pt) + dotstyle); 

label("$A$", (0.6263408942608304,4.2), NE * labelscalefactor); 

dot((0.,0.),linewidth(3.pt) + dotstyle); 

label("$B$", (-0.44658827292841696,0.04763072114368767), NE * labelscalefactor); 

dot((7.,0.),linewidth(3.pt) + dotstyle); 

label("$C$", (7.008893888822507,0.18518574257820614), NE * labelscalefactor); 

dot((1.623345080409327,6.500264738079558),linewidth(3.pt) + dotstyle); 

label("$B_1$", (1.7267810657369815,6.6777827542874775), NE * labelscalefactor); 

dot((-0.46496079380901606,-1.8618150195300045),linewidth(3.pt) + dotstyle); 

label("$A_1$", (-1.1068523758141076,-1.6305405403574376), NE * labelscalefactor); 

dot((10.080396523214976,0.),linewidth(3.pt) + dotstyle); 

label("$B_4$", (10.062615364668826,-0.612633381742001), NE * labelscalefactor); 

dot((-3.0803965232149757,0.),linewidth(3.pt) + dotstyle); 

label("$C_4$", (-3.3077327187664096,-0.612633381742001), NE * labelscalefactor); 

dot((0.007284204967787214,5.552463941947242),linewidth(3.pt) + dotstyle); 

label("$C_1$", (0.1036318128096586,5.714897604245849), NE * labelscalefactor); 

dot((8.151100081632526,-0.9140142233976905),linewidth(3.pt) + dotstyle); 

label("$A_4$", (8.521999124602214,-1.1903644717669786), NE * labelscalefactor); 

dot((-2.308476341169285,1.3881159854868106),linewidth(3.pt) + dotstyle); 

label("$C_3$", (-2.9776006673235647,1.7808239912186203), NE * labelscalefactor); 

dot((-0.7646359770779035,4.164347956460432),linewidth(3.pt) + dotstyle); 

label("$C_2$", (-1.1618743843879151,4.504413415622086), NE * labelscalefactor); 

dot((1.6890544250513695,-1.624864820496926),linewidth(3.pt) + dotstyle); 

label("$A_2$", (1.6167370485893664,-2.125738617521704), NE * labelscalefactor); 

dot((5.997084862772141,-1.150964422430769),linewidth(3.pt) + dotstyle); 

label("$A_3$", (6.211074764502297,-1.603029536070534), NE * labelscalefactor); 

dot((7.966133662513563,1.6250661845198895),linewidth(3.pt) + dotstyle); 

label("$B_3$", (8.081823056011753,1.7808239912186203), NE * labelscalefactor); 

dot((3.7376079411107392,4.8751985535596685),linewidth(3.pt) + dotstyle); 

label("$B_2$", (3.8451283958285725,5.027122497073257), NE * labelscalefactor); 

dot((2.0395968109217,2.660375186246903),linewidth(3.pt) + dotstyle); 

label("$D$", (1.7542920700238853,2.991308179842383), NE * labelscalefactor); 

dot((3.4388364046369224,1.909931693481981),linewidth(3.pt) + dotstyle); 

label("$E$", (3.542507348672631,2.083445038374561), NE * labelscalefactor); 

dot((4.301698851378541,0.8775330211014288),linewidth(3.pt) + dotstyle); 

label("$F$", (4.22,0.93), NE * labelscalefactor); 

dot((2.9561195753832448,0.6030390855677443),linewidth(3.pt) + dotstyle); 

label("$G$", (2.909754250073844,0.10265272971749505), NE * labelscalefactor); 

dot((1.4816619768719694,0.8229159040072803),linewidth(3.pt) + dotstyle); 

label("$H$", (0.9839839499905795,0.43278478116033936), NE * labelscalefactor); 

dot((1.3969966570225139,1.8221911417546572),linewidth(3.pt) + dotstyle); 

label("$I$", (0.9839839499905795,1.8908680083662353), NE * labelscalefactor); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 

 /* end of picture */[/asy]