[asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.4) + fontsize(9); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -5, xmax = 10.231782458472571, ymin = -4, ymax = 4.757055144544911; /* image dimensions */ pen fuqqzz = rgb(0.9568627450980393,0,0.6); pen ffvvqq = rgb(1,0.3333333333333333,0); /* draw figures */ draw(circle((-1.0827599828526435,0.06122686681012432), 3.8484394757927123), linewidth(1)); draw((-2.57427432580244,3.6088840322666065)--(-4.738272658971382,-1.141982788060923), linewidth(1)); draw((-4.738272658971382,-1.141982788060923)--(2.6109521207567776,-1.0190403660952103), linewidth(1)); draw((-2.57427432580244,3.6088840322666065)--(2.6109521207567776,-1.0190403660952103), linewidth(1)); draw(circle((-1.9580411058083014,0.6797534489909157), 1.7749784435623503), linewidth(1)); draw(circle((-2.6241130586807624,2.0690117622326234), 1.5406785866346455), linewidth(1) + fuqqzz); draw((-4.124573881028565,2.4187371229285346)--(-2.9573251984848907,-3.299798520059381), linewidth(1)); draw((-2.9573251984848907,-3.299798520059381)--(1.8784981665217744,2.519160261270455), linewidth(1)); draw((-1.147130075635339,3.9091279687090332)--(-4.124573881028565,2.4187371229285346), linewidth(1) + blue); draw((-1.147130075635339,3.9091279687090332)--(-1.1230378572533952,2.4689486920994947), linewidth(1) + blue); draw((-1.1230378572533952,2.4689486920994947)--(-1.018389890069948,-3.7866742350887836), linewidth(1)); draw((-2.57427432580244,3.6088840322666065)--(-1.147130075635339,3.9091279687090332), linewidth(1) + ffvvqq); draw((-2.57427432580244,3.6088840322666065)--(-1.9580411058083014,0.6797534489909157), linewidth(1) + ffvvqq); draw((-1.9580411058083014,0.6797534489909157)--(-1.018389890069948,-3.7866742350887836), linewidth(1)); draw((-4.124573881028565,2.4187371229285346)--(-1.1230378572533952,2.4689486920994947), linewidth(1) + blue); draw((-1.1230378572533952,2.4689486920994947)--(1.8784981665217744,2.519160261270455), linewidth(1)); draw((-1.147130075635339,3.9091279687090332)--(-1.9580411058083014,0.6797534489909157), linewidth(1) + ffvvqq); draw((-1.9580411058083014,0.6797534489909157)--(-2.9573251984848907,-3.299798520059381), linewidth(1)); /* dots and labels */ dot((-2.57427432580244,3.6088840322666065),dotstyle); label("$B$", (-2.5279841754054746,3.7414818818484883), NE * labelscalefactor); dot((-4.738272658971382,-1.141982788060923),dotstyle); label("$A$", (-4.689332401144001,-1.010880180769387), NE * labelscalefactor); dot((2.6109521207567776,-1.0190403660952103),dotstyle); label("$C$", (2.6670636683877293,-0.8936986504582615), NE * labelscalefactor); dot((-1.018389890069948,-3.7866742350887836),linewidth(4pt) + dotstyle); label("$D$", (-0.9655637712571425,-3.680015037856139), NE * labelscalefactor); dot((-1.9580411058083014,0.6797534489909157),linewidth(4pt) + dotstyle); label("$I$", (-1.9030160137461418,0.7859032840012069), NE * labelscalefactor); dot((-1.9283523117352548,-1.0949766855978653),linewidth(4pt) + dotstyle); label("$E$", (-1.876975673677003,-0.9848398407002481), NE * labelscalefactor); dot((-1.987729899881348,2.4544835835796968),linewidth(4pt) + dotstyle); label("$E'$", (-1.94207652384985,2.5566464087026617), NE * labelscalefactor); dot((-4.124573881028565,2.4187371229285346),linewidth(4pt) + dotstyle); label("$P$", (-4.0773844095192375,2.5175858985989534), NE * labelscalefactor); dot((1.8784981665217744,2.519160261270455),linewidth(4pt) + dotstyle); label("$Q$", (1.9249139764172716,2.6217472588755095), NE * labelscalefactor); dot((-0.07873867433159454,-8.253101919168483),linewidth(4pt) + dotstyle); label("$I_A$", (-9.76719871462608,4.757055144544911), NE * labelscalefactor); dot((-1.147130075635339,3.9091279687090332),linewidth(4pt) + dotstyle); label("$G$", (-1.0957654716028369,4.014905452574449), NE * labelscalefactor); dot((-1.1230378572533952,2.4689486920994947),linewidth(4pt) + dotstyle); label("$M$", (-1.069725131533698,2.5696665787372313), NE * labelscalefactor); dot((-1.063660269107302,-1.0805115770780669),linewidth(4pt) + dotstyle); label("$N$", (-1.0176444513954204,-0.9718196706656785), NE * labelscalefactor); dot((-2.9573251984848907,-3.299798520059381),linewidth(4pt) + dotstyle); label("$H$", (-2.905569106407988,-3.198268746577067), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy] Fun problem, but I can't believe how this is the first question of the day for a Grade 8-9 contest... First we show that $BMIP$ cyclic. By poncelet porism the other tangents from $P,Q$ to the incircle meet at some point $H$ on $(ABC)$, so $GI=GP=GQ$; then $\triangle GBI\cong \triangle GMP$ so $GM=GB$, thus $BMIP$ is an isosceles trapezium. Now by Ptolemy's on $PGCA$, we have $PG\cdot AC=AG(CP-AP)$. Thus we want to show \begin{align*} PG\cdot AC=AG\cdot 2BP&\Leftrightarrow GI\cdot 2AN = AG\cdot 2IN (\text{since} BP=IM=IN, GI=GP)\\ &\Leftrightarrow\frac{GI}{IN}=\frac{AG}{AN}\\ \end{align*} which is true since $\triangle DIN\sim\triangle DGI$ so $\frac{GI}{IN}=\frac{DG}{DI}=\frac{DG}{DA}=\frac{AG}{AN}$ and we are done. Quick commentaryOnce again despite the short solution I found this quite difficult; the condition and result to prove just looks so weird. I had admittedly the help of GGB to hypothesize that $BPIM$ cyclic. Upon first look it should be quite clear that there is most likely some kind of Ptolemy used somewhere; I spent some time playing around with the isosceles trapezium $PQCA$ but got nowhere. The perpendicular bisector of $BC$ should be quite apparent, as it provides symmetry for $PQ$ and $AC$ (and bring in more familiar points like midpoint of arcs). It took me awhile before I noticed $G$ and its importance, but once I saw the poncelet(how is this in a grade 8-9 contest?!) the rest of the problem was quite smooth.