euclidean quantities :
|AB| = euclidean distance between the points A and B.
e(ABC) = angle at B in the euclidean triangle ABC.
standard euclidean theorems
Suppose that A and B lie on a euclidean circle K.
Then the segment AB divides K into two arcs.
Let C and D be points distinct from A and B.
(1) If C and D lie on the same arc, then e(ACB) = e(ADB).
(2) If C and D lie on opposite arcs, then e(ACB)+e(ADB)=π.
(3) If C and D lie on the same side of the line AB, and
e(ACB)=e(ADB), then there is a circle through A,B,C,D.
(4) If C and D lie on opposite sides of the line AB, and
e(ACB)+e(ADB)=π, then there is a circle through A,B,C,D.
Now suppose that AB is not a diameter of K, and that
C and D lie on K.
(5) C,D lie on the same arc if and only if e(ACB)=e(ADB).
This is the major arc if and only if the angles are acute.
(6) C,D lie on opposite arcs if and only if e(ACB)+e(ADB)=π.
Finally, suppose that AB is a diameter of K
(7) C lies
on K if and only if e(ACB)=π/2.
inside K if and only if e(ACB) > π/2.
outside K if and only if e(ACB) < π/2.
We do not offer proofs of these standard results.
There is a Cabrijava applet on the right.
A and B are fixed.
You can vary K by dragging X
You can drag C (on K))
You can drag D freely.
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