Let $T$ be the projection of $B$ onto $AC$, and let $V$ be the $B-$ antipode of the circle centered at $T$ with radius $TB$; $E \in (T)$, and $$\angle BCA = \angle BEA - \angle CBE = \angle TBE - \angle ABE = \angle TVA$$Hence, we see $\angle VDC = 90^{\circ}$, implying that $V,A,D$ are collinear. Additionally, note that $\triangle CBV$ is isosceles with $CT$ the angle bisector of $\angle BCV$. Thus $E$ is the incenter of $\triangle VDC$, and $V,E$ and $O_2$ are collinear. By angle bisector theorem and by $\triangle BAD \sim \triangle VCD$, we see $$\frac{FA}{FD} = \frac{BA}{BD} = \frac{VC}{VD} = \frac{HC}{HD}$$And thus $FH \parallel AC$, with $O$ the circumcenter of quadrilateral $DFEH$. $EG$ is thus the angle bisector of $\triangle BEH$, implying $$\frac{GH}{GB} = \frac{EH}{EB} = \tan \frac{B}{2} \quad \blacksquare$$ [asy][asy] import graph; size(15cm); 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 = -3.6527387539219442, xmax = 25.43884214734359, ymin = -9.601104059952009, ymax = 9.759495076430218; /* image dimensions */ pen qqwwzz = rgb(0,0.4,0.6); pen xdxdff = rgb(0.49019607843137253,0.49019607843137253,1); pen qqzzff = rgb(0,0.6,1); pen yqqqyq = rgb(0.5019607843137255,0,0.5019607843137255); pen qqttzz = rgb(0,0.2,0.6); pen ccwwff = rgb(0.8,0.4,1); pen wwqqcc = rgb(0.4,0,0.8); draw(arc((9.544742712643307,0.4),0.7602329503815035,134.93579947632003,180)--(9.544742712643307,0.4)--cycle, linewidth(1) + xdxdff); /* draw figures */ draw((3.3667454879339376,6.591857783173964)--(6.6,0.4), linewidth(1) + qqwwzz); draw((15.16,0.4)--(6.6,0.4), linewidth(1) + qqwwzz); draw((3.3667454879339376,6.591857783173964)--(15.16,0.4), linewidth(1) + qqwwzz); draw((3.3667454879339376,6.591857783173964)--(9.544742712643307,0.4), linewidth(1) + xdxdff); draw((6.6,0.4)--(7.615297040064838,2.3337744531810367), linewidth(1) + qqzzff); draw((7.615297040064838,2.3337744531810367)--(8.449750894201777,3.9231079012381884), linewidth(1) + qqzzff); draw(circle((9.54804232506882,2.332742743793708), 1.9327455603698553), linewidth(1) + dotted + yqqqyq); draw(circle((12.352371356321655,3.2043360731835677), 3.9682589900561864), linewidth(1) + dotted + yqqqyq); draw((9.544742712643307,0.4)--(9.549771424326064,3.3455598907095863), linewidth(1) + dotted + qqttzz); draw((9.544742712643307,0.4)--(12.352371356321655,3.2043360731835677), linewidth(1) + dotted + qqttzz); draw((3.366745487933938,0.4)--(6.6,0.4), linewidth(1) + dotted + ccwwff); draw((3.366745487933938,0.4)--(3.3667454879339376,6.591857783173964), linewidth(1) + dotted + ccwwff); draw(circle((3.366745487933938,0.4), 6.191857783173964), linewidth(1) + wwqqcc); draw((3.366745487933938,0.4)--(3.3667454879339376,-5.791857783173964), linewidth(1) + dotted + ccwwff); draw((3.3667454879339376,-5.791857783173964)--(15.16,0.4), linewidth(1) + dotted + ccwwff); draw((3.3667454879339376,-5.791857783173964)--(6.6,0.4), linewidth(1) + dotted + qqzzff); /* dots and labels */ dot((3.3667454879339376,6.591857783173964),dotstyle); label("$B$", (3.468109881318138,6.845268766634464), NE * labelscalefactor); dot((6.6,0.4),dotstyle); label("$A$", (6.559723879536252,-0.5036497537200457), NE * labelscalefactor); dot((15.16,0.4),dotstyle); label("$C$", (15.353085005615641,-0.5289908520660958), NE * labelscalefactor); dot((9.544742712643307,0.4),linewidth(4pt) + dotstyle); label("$E$", (9.651337877754367,0.6113585735061557), NE * labelscalefactor); label("$45^{\circ}$", (8.054848681953208,-0.3769442619897956), NE * labelscalefactor,xdxdff); dot((7.615297040064838,2.3337744531810367),linewidth(4pt) + dotstyle); label("$F$", (6.787793764650703,1.8530723924626076), NE * labelscalefactor); dot((8.449750894201777,3.9231079012381884),linewidth(4pt) + dotstyle); label("$D$", (7.902802091876908,4.2604767353373605), NE * labelscalefactor); dot((9.54804232506882,2.332742743793708),linewidth(4pt) + dotstyle); label("$O_1$", (9.676678976100415,1.8277312941165575), NE * labelscalefactor); dot((12.352371356321655,3.2043360731835677),linewidth(4pt) + dotstyle); label("$O_2$", (12.74295187597248,3.3988793915716595), NE * labelscalefactor); dot((9.549771424326064,3.3455598907095863),linewidth(4pt) + dotstyle); label("$G$", (9.651337877754367,3.5509259816479597), NE * labelscalefactor); dot((11.479433537354303,2.3324219684146654),linewidth(4pt) + dotstyle); label("$H$", (11.703966843784425,1.0674983437350565), NE * labelscalefactor); dot((3.366745487933938,0.4),linewidth(4pt) + dotstyle); label("$T$", (3.468109881318138,0.6113585735061557), NE * labelscalefactor); dot((3.3667454879339376,-5.791857783173964),linewidth(4pt) + dotstyle); label("$V$", (3.0373112094352863,-6.864265438578604), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy][/asy]