Show 2 of 3 points on a circle are the diameter

trollcast

1. The problem statement, all variables and given/known data
Points A B and C lie on the circumference of a circle where
$$A =(-3,2)\\B=(-1,6)\\C=(7,2)$$

Show that AC is the diameter of the circle.

2. Relevant equations

3. The attempt at a solution

Would it be sufficient to show that the angle ABC is a right angle and therefore by the inscribed angle theorem that any angle subtended from a diameter is a right angle?

Mandelbroth

Consider a point O, such that O is the midpoint of AC. If AC is the diameter of the circle, then any point on the circumference of the circle should be equidistant from point O.

I think you can finish from there.

trollcast

Thanks I used a similar solution by finding a mid point of AC and then showing that OB was the same length as OC?

So is the diameter not the only 2 points you can subtend right angle triangle from then? (I can't picture any other cases)

Mandelbroth

That's right.

Naw. I just find it simpler to use less theorems.

haruspex

Not exactly. You are trying to show a certain length is a diameter, so you would need a theorem that says the only chords which subtend a right angle on the circumference are diameters. In fact, I believe the inscribed angle theorem says that, but that's not how you quoted it.

symbolipoint

Coordinate geometry for finding equation for the circle which contains those three points could be easier to handle. The resulting and simplified equation will show the size of the radius; then, you merely use distance formula to show that the intended given points (giving length, AC) are twice the size of radius.

Mandelbroth

...was my answer wrong?

symbolipoint

No. In fact, maybe a better method is to understand that if two of the points are the endpoints of a diameter, then the three points given, being all on the circle, form a right-triangle. The longer side would be the hypotenuse, and therefore the circle's diameter.

My earlier posted method of coordinate geometry to try to make a system of three equations would not be very neat to solve.

symbolipoint

Your idea now seems a correct way. You can also find that the slopes of two of the sides are opposite-reciprocals of each other (meaning, intersect at right-angle). You can find the midpoint of the longest side and use this to fill-in the standard equation of a circle and show that the three points satisfy the equation.

EDIT: Just having solved this most of the way through, what I suggest is very good; but not necessary to check the slopes of any segments. You can if you wish.
IF longest side is diameter, then its midpoint is the center of the circle. Longest side length indicates the diameter from which you get the size of radius. You use the found center point of this longest side to fill-in the standard form of equation for a circle. NOW, you can check to see if each point satisfies the equation.

Mentallic

I'm not sure why everyone has been leading the OP away from his solution. Showing angle ABC to be a right angle is an ingenious solution, considering it's very simple to do and it's short and sweet.

haruspex

For my part, because the OP ends thus: "Would it be sufficient to show that the angle ABC is a right angle and therefore by the inscribed angle theorem that any angle subtended from a diameter is a right angle?"
The logic of that last part is backwards from what's needed here, as in "A implies B" versus "B implies A". It just has to be changed to "any chord that subtends a right angle at the circumference is a diameter" (that is, assuming that the 'inscribed angle theorem' says that).

Mentallic

I felt like the OP was just giving us a summary of his reasoning, merely asking a question to see if he was on the right track, but I agree with you that it's very easy to make that logical mistake that you pointed out and it's good to see that you're explicitly critiquing it.

Oh, and it's not your part I was particularly concerned with

But I realize now it was a bad choice of words to say that "everyone" has been leading the OP away...

trollcast

Thanks,

I think I maybe got the circle theorem wrong in my OP

http://en.wikipedia.org/wiki/Thales%27_theorem

Is that basically what I was talking about showing ABC to be right angled would be sufficient to show the AC is the diameter?

haruspex

The result you need is the one described as the converse at that link.