Question 2935: Is there a formula for finding the radius of an arc, when only 3 points are known on the arc?
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! To get the answer you should supply clearly what the given conditions are.
That is, the coordinates of the three points, the lengths of the 3 chords
formed by these 3 points, etc. Anyway, your question is not clear.
Forgiven 3 points A,B,C.
It seems you want to find the radius R of the circumscribed circle of
triangle ABC .
Start from the area of ABC, we have
area of ABC = 1/2 bc sinA (a 1/2 height base)
the use thelaw of sine (where R is the radius of the circumscribed circle of triangle ABC)
a/sin A = b/sin B =c/ sin C = 2R.
to replace sin A by a/(2R)
hence,we obtain area of ABC = 1/2 bc sinA = 1/2 bc *a/(2R)
= abc/(4R)
Thus,we get a simple formula to find the radius R.
That is, R = abc/ (4 area of ABC)
What remains is to find the formula area of ABC.
In fact, if the three sides a,b,c are given,
I hope you know
area of ABC = sqrt(s(s-a)(s-b)(s-c))
where s = (a+b+c)/2.
A better way, using determinant,
if given A(x1,y1) ,B(x2,y2) ,C(x3,y3)
then the area of ABC = 1/2 times
|1 x1 y1|
|1 x2 y2| (with absolute value)
|1 x3 y3|
If you need, you can the length of the 3 sides a,b,c by
the basic distance formula. But,I am lazy to type here.
Kenny
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