Powers $P_A,P_B,P_C$ of $A,B,C$ WRT 9-point circle $\odot(DEF)$ are $P_A= \frac{_1}{^2} AB \cdot AC \cdot \cos A \ , \ P_B= \frac{_1}{^2} BC \cdot BA \cdot \cos B \ , \ P_C= \frac{_1}{^2} CA \cdot CB \cdot \cos C $ Lengths of altitudes $AD,CE,CF$ are given by $AD=\frac{AB \cdot AC}{2R} \ \ , \ \ \ BE=\frac{BC \cdot BA}{2R} \ \ , \ \ CF=\frac{CA \cdot CB}{2R}$ Substituting the latter expressions into the proposed inequality yields $\cos A+ \cos B + \cos C \le \frac{3}{2} \ ,$ which follows from Jensen's inequality for $\cos \theta \ , (0, \frac{_{\pi}}{^{2}})$