If a points 
,
, and 
are marked on each side of a triangle 
, one on each side (or on a side's
extension), then the three Miquel circles (each
through a polygon vertex and the two marked points
on the adjacent sides) are concurrent at a point 
called the Miquel
point. This result is a slight generalization of the so-called pivot
theorem.
If 
lies in the interior of the triangle, then it satisfies
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
The lines from the Miquel point to the marked points make equal angles with the respective sides. (This is a by-product of the Miquel equation.)
A generalized version of Miquel's theorem states that given four lines 
, ..., 
each intersecting the other
three, the four Miquel circles passing through
each subset of three intersection points of the lines meet in a point known as the
4-Miquel point 
.
Furthermore, the centers of these four Miquel circles
lie on a circle 
(Johnson 1929, p. 139). The lines from 
to given points on the sides make equal angles
with respect to the sides.
Moreover, given 
lines taken by 
s
yield 
Miquel circles like 
passing through a point 
, and their centers lie on a circle 
.
