GeoMetrix wrote:

Nice and easy.
Define $\omega=\odot(CIQ)$ and let $H=\omega \cap \overline{BC}$ . Observe that trivially we have that $N \in \overline{AC}$ (just consider the line through $I$ parallel to $BC$ ). Now since $\angle NCH=\angle NHC=\angle ABC$ $\implies$ $\overline{NH} \| \overline{AB}$ . Hence by the converse of ceva's theorem we have that $\overline{AH},\overline{BN},\overline{CM}$ are concurrent. Now define $K= \overline{AD} \cap \overline{BC}$ . Observe that to show that $K \in \overline{MN}$ we need to show that $(B,C;H,K)=-1$ . But if $\overline{DH} \cap \omega=J$ then we have that $(B,C;H,K)\overset{D}{=}(A,J;B,C)$ . Hence we just need to show that $(AJBC)$ is harmonic. For this notice that by reims theorem we have that $\overline{AJ} \| \overline{QH}$ . But as $\angle QHC=90^\circ$ we have that $\overline{AJ} \perp \overline{BC}$ by which we have that $J$ is the midpoint of minor $\widehat{BC}$ . Done $\blacksquare$ .

$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(14cm); 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 = -10.006881897432205, xmax = 25.634412731501875, ymin = -8.577795838543958, ymax = 14.83226710342647; /* image dimensions */ pen ffqqff = rgb(1,0,1); pen wvvxds = rgb(0.396078431372549,0.3411764705882353,0.8235294117647058); pen dtsfsf = rgb(0.8274509803921568,0.1843137254901961,0.1843137254901961); pen qqffff = rgb(0,1,1); pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); /* draw figures */ draw((1.5534973808986277,9.160392608982256)--(-3.5857910956748746,-1.8868970241407366), linewidth(0.8)); draw((1.5534973808986277,9.160392608982256)--(7.796798634472622,-1.3026893337662244), linewidth(0.8)); draw((-3.5857910956748746,-1.8868970241407366)--(7.796798634472622,-1.3026893337662244), linewidth(0.8)); draw((1.5534973808986277,9.160392608982256)--(1.9295955015644004,1.832569513956508), linewidth(0.4) + ffqqff); draw(circle((5.807241390273251,2.031588419358404), 3.8827498197705226), linewidth(0.4) + wvvxds); draw(circle((1.9067865071150452,2.276975553541132), 6.892477317034895), linewidth(0.4) + dtsfsf); draw(circle((3.5081332893469326,4.891548128823499), 4.6950649335634775), linewidth(0.4) + wvvxds); draw((8.19899262334367,5.090227365365952)--(5.807241390273251,2.031588419358404), linewidth(0.4) + qqffff); draw((3.8176841460738786,5.365866172483034)--(8.19899262334367,5.090227365365952), linewidth(0.4) + qqffff); draw((8.19899262334367,5.090227365365952)--(7.796798634472622,-1.3026893337662244), linewidth(0.4) + wrwrwr); draw((3.8176841460738786,5.365866172483034)--(1.9295955015644004,1.832569513956508), linewidth(0.4) + ffqqff); draw((1.9295955015644004,1.832569513956508)--(7.796798634472622,-1.3026893337662244), linewidth(0.4) + qqffff); draw((3.8176841460738786,5.365866172483034)--(4.169500681742824,-1.488859213336944), linewidth(0.4) + qqffff); draw((5.807241390273251,2.031588419358404)--(4.169500681742824,-1.488859213336944), linewidth(0.4) + qqffff); draw((1.9295955015644004,1.832569513956508)--(2.2600756333314593,-4.606441501899994), linewidth(0.4) + ffqqff); draw((2.2600756333314593,-4.606441501899994)--(8.19899262334367,5.090227365365952), linewidth(0.4) + green); draw((-1.0161468573881236,3.63674779242076)--(17.798762343332783,-0.7893418218123583), linewidth(0.4) + linetype("4 4") + blue); draw((1.5534973808986277,9.160392608982256)--(17.798762343332783,-0.7893418218123583), linewidth(0.4) + linetype("4 4") + blue); draw((1.9295955015644004,1.832569513956508)--(5.807241390273251,2.031588419358404), linewidth(0.4) + wrwrwr); draw((7.796798634472622,-1.3026893337662244)--(17.798762343332783,-0.7893418218123583), linewidth(0.4) + linetype("4 4") + blue); /* dots and labels */ dot((1.5534973808986277,9.160392608982256),dotstyle); label("$A$", (1.6396828828887742,9.383171533517269), NE * labelscalefactor); dot((-3.5857910956748746,-1.8868970241407366),dotstyle); label("$B$", (-3.4819992193005715,-1.6553396638953617), NE * labelscalefactor); dot((7.796798634472622,-1.3026893337662244),dotstyle); label("$C$", (7.88392544583195,-1.0706727572527435), NE * labelscalefactor); dot((1.9295955015644004,1.832569513956508),linewidth(4pt) + dotstyle); label("$I$", (2.0138697031400508,2.01636850982028), NE * labelscalefactor); dot((5.807241390273251,2.031588419358404),linewidth(4pt) + dotstyle); label("$N$", (5.896057963247044,2.2268485962116227), NE * labelscalefactor); dot((3.8176841460738786,5.365866172483034),linewidth(4pt) + dotstyle); label("$Q$", (3.9081904806621375,5.547756625941694), NE * labelscalefactor); dot((8.19899262334367,5.090227365365952),linewidth(4pt) + dotstyle); label("D", (8.28149894234893,5.267116510753237), NE * labelscalefactor); dot((-1.0161468573881236,3.63674779242076),linewidth(4pt) + dotstyle); label("$M$", (-0.9328515063387511,3.8171425822795437), NE * labelscalefactor); dot((17.798762343332783,-0.7893418218123583),linewidth(4pt) + dotstyle); label("Q", (17.893422887553594,-0.602939231938649), NE * labelscalefactor); dot((4.675148007685625,3.928851637608016),linewidth(4pt) + dotstyle); label("$F$", (4.773497502493214,4.121169373733705), NE * labelscalefactor); dot((4.169500681742824,-1.488859213336944),linewidth(4pt) + dotstyle); label("$H$", (4.258990624647709,-1.3045395199097907), NE * labelscalefactor); dot((2.2600756333314593,-4.606441501899994),linewidth(4pt) + dotstyle); label("$J$", (2.3646698471256222,-4.414967463248519), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$

When you use Geogebra-Asymptote conversion, how did you doit. Why did mine have an error?