First, $G$ be point on ray $CA$ such that $AG = AF$, where $F$ is the midpoint of $AB$ Notice that $$GA^2 = \frac{AB^2}{4} = \frac{AB}{2} \cdot \frac{AD}{\sin \angle C} = \frac{AO}{2 \sin \angle C} \cdot \frac{AD}{\sin \angle C} = AD \cdot AM$$Thus, $(GDM)$ is tangent to $AC$, so it leave to show that $\square GDMP$ is cyclic. Observe that $2AG = AB$ and $\angle GAB = 60^{\circ}$ is a $60^{\circ}$ right triangle, so $\angle BGA = 90^{\circ} = 180^{\circ} - \angle ADB$, so we have $GADB$ cyclic with center $F$. So, we have $\angle BDA = 30^{\circ}$ Note that $BFDOP$ is cyclic. Consider the following angle chasing $$\angle FPB = \angle FDB = 90^{\circ} - \angle BAO = \angle ACB = 60^{\circ} - \angle ABC$$Also, $$BOP = 60^{\circ} \rightarrow 2PO = BO = AO = 2MO \rightarrow PO = MO$$So, we have $$2\angle MPO = \angle AOC = 2\angle ABC \rightarrow \angle MPO = \angle ABC$$$$\angle BPM = 90^{\circ} - \angle FPB - \angle MPO = 90^{\circ} - 60^{\circ} + \angle ABC - \angle ABC = 30^{\circ} = \angle GDA$$which yields $GDMP$ cyclic. [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(0.3) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -9.915368734106012, xmax = 8.39664650997251, ymin = -9.611564390036216, ymax = 5.6292049177719035; /* image dimensions */ pen ccqqqq = rgb(0.8,0,0); /* draw figures */ draw((-0.91,2.15)--(-3.45,0.09), linewidth(1)); draw((-0.91,2.15)--(5.270663057950133,-0.21722997968993094), linewidth(1)); draw((5.270663057950133,-0.21722997968993094)--(-3.45,0.09), linewidth(1)); draw(circle((0.8216418732365139,-2.581053571854731), 5.038000742050905), linewidth(1)); draw(circle((-3.795695821636521,-0.8125863192035285), 3.7987311136407627), linewidth(1)); draw((-0.044179063381743044,-0.21552678592736552)--(-1.4028687191202978,-3.7629653679371318), linewidth(1)); draw((-2.4370061658979685,2.734852262806236)--(-0.91,2.15), linewidth(1)); draw((-2.4370061658979685,2.734852262806236)--(-3.45,0.09), linewidth(1)); draw((-0.044179063381743044,-0.21552678592736552)--(0.8216418732365139,-2.581053571854731), linewidth(1)); draw((0.8216418732365139,-2.581053571854731)--(-1.4028687191202978,-3.7629653679371318), linewidth(1)); draw((-2.4370061658979685,2.734852262806236)--(-0.5451616025684995,1.1532176802863074), linewidth(1)); draw((-0.5451616025684995,1.1532176802863074)--(-0.044179063381743044,-0.21552678592736552), linewidth(1)); draw((-1.4028687191202978,-3.7629653679371318)--(-2.4370061658979685,2.734852262806236), linewidth(1)); draw((-0.91,2.15)--(-0.5451616025684995,1.1532176802863074), linewidth(1)); draw((-3.45,0.09)--(-1.4028687191202978,-3.7629653679371318), linewidth(1)); draw((-3.45,0.09)--(-0.5451616025684995,1.1532176802863074), linewidth(1)); draw((0.8216418732365139,-2.581053571854731)--(5.270663057950133,-0.21722997968993094), linewidth(1)); draw(circle((-1.3141790633817434,-1.245526785927365), 2.5190003710254527), linewidth(1) + linetype("4 4") + ccqqqq); /* dots and labels */ dot((-0.91,2.15),dotstyle); label("$A$", (-0.817091298838752,2.3732223838310778), NE * labelscalefactor); dot((-3.45,0.09),dotstyle); label("$B$", (-3.357219516806769,0.31802773474786156), NE * labelscalefactor); dot((5.270663057950133,-0.21722997968993094),dotstyle); label("$C$", (5.37158472311969,0.017830763533459183), NE * labelscalefactor); dot((0.8216418732365139,-2.581053571854731),linewidth(4pt) + dotstyle); label("$O$", (0.9148143043212595,-2.40683708089056), NE * labelscalefactor); dot((-0.5451616025684995,1.1532176802863074),linewidth(4pt) + dotstyle); label("$D$", (-0.44761810349794956,1.3340790219350696), NE * labelscalefactor); dot((-0.044179063381743044,-0.21552678592736552),linewidth(4pt) + dotstyle); label("$M$", (0.037315465386853715,-0.028353385884141182), NE * labelscalefactor); dot((-1.4028687191202978,-3.7629653679371318),linewidth(4pt) + dotstyle); label("$P$", (-1.3020248677235553,-3.5845328910393692), NE * labelscalefactor); dot((-2.4370061658979685,2.734852262806236),linewidth(4pt) + dotstyle); label("$G$", (-2.341168229619562,2.927432176842282), NE * labelscalefactor); dot((-2.18,1.12),linewidth(4pt) + dotstyle); label("$F$", (-2.0871554078227605,1.3109869472262694), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]