I am struggling with finding the equation of the nine-point circle in (1). The "c" in the vertices are confusing me. I have found the midpoints of AB, AC, and BC in terms of "c", but in order to find the equation I need to get either three points on that circle or the coordinates of the center of the circle.
In terms of "c", I found the following midpoints:
$M_{BC} = (-2c+2,\frac{-c}{2})$
$M_{AB} = (-2c+2,0)$
$M_{AC} = (-4, \frac {-c}{2})$
I also found, what I think should be the center of the circle in terms of "c":
$U=(-c+3, \frac{-c}{4})$
I don't know if I am correct. Can someone please assist me with some guidance on how to get the equation for the nine-point circle?
Welcome to MSE.
Hint:As you see in figure the triangle is rightangled at A.
1-A is where three altituds intersect. O is the center of circumcircle and it is on mid point of BC.The center of nine point circle N is the mid point of segment connecting A and O.
2- The measure of radius of nine point circle is half of the radius of circumcircle , so $r=NA=\frac{BC}4$
3- the equation of circle is:
$ (x-x_O)^2+(y-y_O)^2=r^2$
4- the figure shows that two circles are tangent because their centers are collinear and also they both cross vertex A.
I think these are enough data to answer all questions.