Mathematics - Circle Question with Solution | TestHub

The circumcircles of $△ABC$ and $△DEF$ (nine-point circle) touch if and only if the triangle $ABC$ is right-angled.

If $△ABC$ is right-angled, say at A, then $\cos A=0$, $\cos B\ne 0$, $\cos C\ne 0$.

Then $\sum {\cos }^{2}A={\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=0+{\cos }^{2}B+{\cos }^{2}C$.

Since $A=90^{\circ }$, $B+C=90^{\circ }$, so $C=90^{\circ }-B$.

Thus, ${\cos }^{2}B+{\cos }^2(90^{\circ }-B)={\cos }^{2}B+{\sin }^{2}B=1$.

So, $\sum {\cos }^{2}A=1$.

The greatest integer less than or equal to 1 is $[1]=1$.