$\displaystyle \begin{array}{{>{\displaystyle}l}} \mathrm{a.\ AH\ đi\ qua\ A( 1;4) \ nhận\ \overrightarrow{BC}( 3;1) \ là\ vtpt}\\ \mathrm{AH:3( x-1) +y-4=0\ }\\ \mathrm{hay\ AH:\ 3x+y-7=0}\\ \mathrm{b.\ Gọi\ I\left(\frac{7}{2} ;3\right) \ là\ trung\ \ điểm\ AC\ cũng\ chính\ là\ tâm\ đường\ tròn\ ( C)}\\ \mathrm{R=\frac{AC}{2} =\frac{\sqrt{( 6-1)^{2} +( 2-4)^{2}}}{2} =\frac{\sqrt{29}}{2}}\\ \mathrm{Vậy\ ( C) :\ \left( x-\frac{7}{2}\right)^{2} +( y-3)^{2} =\frac{29}{4}} \end{array}$