Problem 14: The third of the parallel lines

by henderson, May 31, 2016, 5:52 PM

$\bf\color{red}Problem\ 14\$ Given is a triangle $ABC$ inscribed in circle $c$ . Let $D,E$ be points on the segments $AB,AC$ respectively such
that $DE\parallel BC$ . We consider the tangents $\varepsilon _1,\varepsilon _2$ of $c$ at $C,B$ respectively. Let $S\equiv BE\cap \varepsilon _1$ and $T\equiv CD\cap \varepsilon _2$ . Also,let $M\equiv TA\cap c$ and $N\equiv SA\cap c$ with $M,N\neq A$ . Prove that $MN\parallel BC$ .
$[asy]import graph; size(9cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.319332900674542, xmax = 12.46238917406057, ymin = -3.2834042311679315, ymax = 6.211517469011155; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); pen qqwuqq = rgb(0.,0.39215686274509803,0.); draw((0.3989981582330802,2.5552050251389775)--(0.,0.)--(4.,0.)--cycle, aqaqaq); /* draw figures */ draw((0.3989981582330802,2.5552050251389775)--(0.,0.), uququq); draw((0.,0.)--(4.,0.), uququq); draw((4.,0.)--(0.3989981582330802,2.5552050251389775), uququq); draw(circle((2.,0.9964522548556711), 2.234483630776236)); draw((0.,0.)--(-0.8529264345008228,1.7119263473880404)); draw((-0.8529264345008228,1.7119263473880404)--(4.,0.)); draw((0.,0.)--(5.839458241941738,3.692014811503779)); draw((5.839458241941738,3.692014811503779)--(4.,0.)); draw((-0.8529264345008228,1.7119263473880404)--(1.1571205626701906,3.0658655413534492)); draw((0.3989981582330802,2.5552050251389775)--(5.839458241941738,3.692014811503779)); draw((1.1571205626701906,3.0658655413534492)--(2.8428794373298087,3.0658655413534492), linewidth(1.6) + red); draw((0.20883313408761875,1.3373782876573805)--(2.115258216224203,1.3373782876573805), linewidth(1.6) + qqwuqq); /* dots and labels */ dot((0.3989981582330802,2.5552050251389775),linewidth(3.pt) + dotstyle); label("$A$", (0.15,2.710067938380329), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.4,-0.03431142346545344), NE * labelscalefactor); dot((4.,0.),linewidth(3.pt) + dotstyle); label("$C$", (4.134617317425102,-0.0027668330994099592), NE * labelscalefactor); dot((0.20883313408761875,1.3373782876573805),linewidth(3.pt) + dotstyle); label("$D$", (-0.1,1.5271457996536988), NE * labelscalefactor); dot((2.115258216224203,1.3373782876573805),linewidth(3.pt) + dotstyle); label("$E$", (2.084218943632278,1.4956012092876552), NE * labelscalefactor); dot((-0.8529264345008228,1.7119263473880404),linewidth(3.pt) + dotstyle); label("$T$", (-1.0860123881550887,1.7952748177650684), NE * labelscalefactor); dot((5.839458241941738,3.692014811503779),linewidth(3.pt) + dotstyle); label("$S$", (5.901114377923536,3.7825840108258073), NE * labelscalefactor); dot((1.1571205626701906,3.0658655413534492),linewidth(3.pt) + dotstyle); label("$M$", (1.106336642284931,3.2147813842370248), NE * labelscalefactor); dot((2.8428794373298087,3.0658655413534492),linewidth(3.pt) + dotstyle); label("$N$", (2.9043782931494078,3.1674644986879597), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]$

$\blacktriangleright$ Link to the problem

This post has been edited 6 times. Last edited by henderson, Sep 11, 2016, 7:54 PM

Parallel Lines

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Problem 14: The third of the parallel lines

by henderson, May 31, 2016, 5:52 PM

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