Toss $\odot(ABC)$ onto the unit circle, then \begin{align*}h&=a+b+c\\d&=\frac{ab-bc+ca}{a}\\e&=\frac{ab+bc-ca}{b}\\f&=\frac{bc-ab+ca}{c}\end{align*}For $\overline{D-E-F}$ we must have $\frac{d-e}{e-f}=\overline{\left(\frac{d-e}{d-f}\right)}$ which is equivalent to proving
\begin{align*}
\frac{d-e}{d-f}=\frac{c(b-a)(ab-bc-ca)}{b(c-a)(ac-bc-ab)}=\frac{(a-b)(c-a-b)}{(a-c)(b-a-c)}=\overline{\left(\frac{d-e}{d-f}\right)}
&\iff c(ab-bc-ca)(b-a-c)=b(ac-bc-ab)(c-a-b)\\
&\iff (a+b+c)(ab+bc+ca)=4abc \\&\iff h\overline{h}=4\\&\iff |h|=2
\end{align*}and thus $OH=2=2R$ and we are done!
[asy][asy]
import graph; size(10cm);
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 = -7.694081287150866, xmax = 0.4768291617537498, ymin = -2.0950501157930743, ymax = 5.236581455833256; /* image dimensions */
pen zzttqq = rgb(0.6,0.2,0);
draw((-3,4)--(-3.844974712047039,0.7797212109761972)--(-3.33,0.2)--cycle, linewidth(0.6) + zzttqq);
/* draw figures */
draw((-3,4)--(-3.844974712047039,0.7797212109761972), linewidth(0.6));
draw((-3.844974712047039,0.7797212109761972)--(-3.33,0.2), linewidth(0.6));
draw((-3.33,0.2)--(-3,4), linewidth(0.6));
draw(circle((-1.898703000807973,1.9900321027017454), 2.2919044545091602), linewidth(0.6));
draw((-7.142406349521041,0.32024070081159844)--(-2.7227998066557872,0.682269179718536), linewidth(0.6));
draw((-6.377568710431095,0.9996570055727055)--(-1.8987030008079708,1.9900321027017447), linewidth(0.6));
/* dots and labels */
dot((-3,4),linewidth(3pt) + dotstyle);
label("$A$", (-3.160045973118789,4.088602301854922), NE * labelscalefactor);
dot((-3.844974712047039,0.7797212109761972),linewidth(3pt) + dotstyle);
label("$B$", (-4.134381221453342,0.7218399090949367), NE * labelscalefactor);
dot((-3.33,0.2),linewidth(3pt) + dotstyle);
label("$C$", (-3.4205118315844616,-0.020970131714573053), NE * labelscalefactor);
dot((-6.377568710431095,0.9996570055727055),linewidth(3pt) + dotstyle);
label("$H$", (-6.555749016819409,1.0884214876762532), NE * labelscalefactor);
dot((-1.8987030008079708,1.9900321027017447),linewidth(3pt) + dotstyle);
label("$O$", (-1.8577166807904262,2.0434629687170514), NE * labelscalefactor);
dot((-7.142406349521041,0.32024070081159844),linewidth(3pt) + dotstyle);
label("$D$", (-7.105621384691384,0.3745520978073737), NE * labelscalefactor);
dot((-2.7227998066557872,0.682269179718536),linewidth(3pt) + dotstyle);
label("$E$", (-2.59087983795306,0.6253710726261692), NE * labelscalefactor);
dot((-4.578237152753622,0.5275271794198101),linewidth(3pt) + dotstyle);
label("$F$", (-4.64566605473781,0.6060773053324157), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy][/asy]
@below I have gotten rusty at bashing and was getting bored, so just thought of doing this problem