and 
intersect at 
and 
.
be the circumcircle of the triangle 
;
,
and 
intersects 
.
on 
.
and 
meet at 
,
.
.
Y by Adventure10, Mango247
Let 
be a triangle, let 
be the midpoint of the side 
, and let 
be the orthogonal projection of 
on the line , similarly, define 
and 
. The lines 
and 
meet at 
, and the tangent of the circle at meets the line at T
, similarly, define 
and 
. Show that the perpendiculars through 
to the lines 
, respectively, are concurrent.
Flavian Georgescu
be a triangle, let 
be the midpoint of the side 
, and let 
be the orthogonal projection of 
on the line , similarly, define 
and 
. The lines 
and 
meet at 
, and the tangent of the circle at meets the line at T
, similarly, define 
and 
. Show that the perpendiculars through 
to the lines 
, respectively, are concurrent.Flavian Georgescu
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(20.828545285025918cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -2.7662749585150563, xmax = 2.4408613627414235, ymin = -1.0869284944439497, ymax = 1.3535319681780489; /* image dimensions */
pen evevev = rgb(0.8980392156862745,0.8980392156862745,0.8980392156862745);
pair A = (-0.5868660024523246,0.809684071206559), B = (-0.9216716155609753,-0.3879709178137217), C = (0.9033689369585133,-0.42886427193046256), M_C = (-0.7542688090066499,0.21085657669641866), M_B = (0.15825146725309436,0.19040989963804825), P_B = (-0.19616392707689714,0.4849678836316376), P_C = (-0.7999881728521875,0.04731085923954909), S_A = (-0.5797436060494064,0.20694602198738823), L_A = (-1.2127622086043015,-0.014330890846111746), T_A = (-2.1997300281063414,-0.35933369374001767);
filldraw(A--B--C--cycle, evevev, linewidth(0.8));
/* draw figures */
draw(circle((0.,0.), 1.), linewidth(0.4)); /* special point *//* special point *//* special point */
draw(A--B, linewidth(0.4));
draw(B--C, linewidth(0.4));
draw(C--A, linewidth(0.4)); /* special point *//* special point *//* special point */
draw(B--P_B, linewidth(0.4));
draw(C--P_C, linewidth(0.4));
draw(circle((-0.3025843405273932,-0.003575559268812891), 0.5), linewidth(0.4));
draw(P_B--P_C, linewidth(0.4));
draw(M_B--M_C, linewidth(0.4)); /* special point */
draw(P_B--L_A, linewidth(0.4));
draw(M_B--L_A, linewidth(0.4));
draw((xmin, 0.3495583098828616*xmin + 0.40960021708341426)--(xmax, 0.3495583098828616*xmax + 0.40960021708341426), linewidth(0.4)); /* line *//* special point *//* special point *//* special point */
draw(B--T_A, linewidth(0.4));
draw(A--T_A, linewidth(0.4));
draw((xmin, -2.8607530467094455*xmin-0.8691946333191213)--(xmax, -2.8607530467094455*xmax-0.8691946333191213), linewidth(0.4) + linetype("4 4")); /* line */
/* dots and labels */dot(A,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$A$", (-0.5843248097143211,0.842666348007492), NE * labelscalefactor); dot(B,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$B$", (-0.9874777991260225,-0.4775827547224417), NE * labelscalefactor); dot(C,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$C$", (0.9544270582692722,-0.4775827547224417), NE * labelscalefactor); dot(M_C,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$M_C$", (-0.8520676347434664,0.2887156755333939), NE * labelscalefactor); dot(M_B,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$M_B$", (0.16966360559763963,0.20870057839824638), NE * labelscalefactor);
dot(P_B,linewidth(4.pt) + dotstyle);
label("$P_B$", (-0.15962929415084934,0.5472259893546396), NE * labelscalefactor);
dot(P_C,linewidth(4.pt) + dotstyle);
label("$P_C$", (-0.9259277244066788,-0.05288723915896658), NE * labelscalefactor); dot(S_A,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$S_A$", (-0.5904798171862554,0.1317629849990661), NE * labelscalefactor); dot(L_A,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$L_A$", (-1.2706081428350038,0.04867038412795139), NE * labelscalefactor); dot((-0.3025843405273932,-0.003575559268812891),linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$N$", (-0.2519544062298649,0.002507828088443222), NE * labelscalefactor); dot((-0.3983398207705171,0.27035722257583006),linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$A'$", (-0.3935195780843555,0.2948706830053283), NE * labelscalefactor); dot(T_A,linewidth(0.75pt) + dotstyle + invisible,UnFill(0));
label("$T_A$", (-2.2892618794401423,-0.4991252808742122), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/e/0/0/e0035c9ab35f453b2776848100a95b0dff86c557.png)
and 
be the foot of perpendicular from 
.
wrt 
is the 
also 
and 
are polars of 
and 
wrt 
are concurrent at 
is the orthocenter of 
,
passes through 
,
and 
pass through th nine-point center so 
and 
