Sorry in advance for the long post; I am doing a set of questions on finding the equation of a circle given 3 points.
The first set of points are (0,0), (2,0), and (4,-2).
There are a few ways I can see of doing this question but I can't help feeling there is something I am meant to spot that I haven't yet - possibly to do with circle theorems.
The obvious idea is to create 3 simultaneous equations and solve them for a, b, and r, where (a,b) is the centre of the circle, and r is the radius.
But this seems really laborious.
The second idea I had is to somehow use the circle theorem that the angle subtended at the circumference is half that at the centre, but I couldn't see an obvious way in.
The way I ended up solving it was by considering the symmetry of a circle;
If (0,0) and (2,0) form a chord of the circle which is parallel to the x-axis, then the centre of the circle must have an x-coordinate in line with the midpoint of the chord.
So the x-coordinate of the centre is 1.
Then to continue the question, if you consider symmetry again, there must be a coordinate at (-2,-2).
And then just by looking at these coordinates, by symmetry again there is another coordinate at (-2, -4) and (4, -4). At this point, looking at the sketch (with the help of Desmos), you can see that the centre's y-coordinate is at -3.
This second bit of reasoning to find the y-coordinate is a bit unsatisfactory and I can't fully explain it.
Can anyone suggest a neater way of going about this problem?
And/or explain why my method was good or bad?
P.S. The rest of the set of questions were:
- (2,2), (4,3), & (6,9)
- (-1,1), (2,-1), & (-2,0)
- (0,0), (a,0, & (1,1)