Let $u=(6,7), v=(-2, 3), w=(7,0)$. Find the center and the radius of the circumscribed circle of triangle $[u,v,w]$.
My approach would look like this:
(1) Determine the angles of the triangle $[u,v,w]$ via dot products.
(2) If we draw three lines from the the center $o$ of the circle to the three vertices, the triangle $[u,v,w]$ will be made of three equilateral triangles. We can then easily find from (1) all the nine angles of the three triangles.
(3) Determine the heights of the three inscribed triangles, which will be the same for all of the three, and which will be the radius $r$ of the inscribed circle.
(4) Then we can drop a perpendicular line from $o$ to, say, $[v,w]$ in order to find the coordinates of $o$.
But is there a much less tedious way to solve this problem? And do you think my approach is correct?
Best Answer
HINT:
If $(h,k)$ is the center with radius $=r$
$$r^2=(h-6)^2+(k-7)^2=(h+2)^2+(k-3)^2=(h-7)^2+(k-0)^2$$
$$(h-6)^2+(k-7)^2=(h+2)^2+(k-3)^2\ \ \ \ (1)$$
and $$(h+2)^2+(k-3)^2=(h-7)^2+(k-0)^2\ \ \ \ (2)$$
will give two simultaneous equations in $h,k$