The reflection triangle is the triangle 
obtained by reflecting
the vertices of a reference
triangle 
about the opposite sides (Grinberg 2003). It is perspective
to the reference triangle with the orthocenter 
as the perspector,
and has trilinear vertex matrix
![[-1 2cosC 2cosB; 2cosC -1 2cosA; 2cosB 2cosA -1].](/images/equations/ReflectionTriangle/NumberedEquation1.svg) |
(1)
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The triangles 
, 
, and 
are congruent to the
reference triangle. Their circumcircles
pass through its orthocenter 
and form its Johnson circles
(F. M. Jackson, pers. comm., Feb. 13, 2010).
If the reference triangle is an acute triangle, then the quadrilaterals 
, 
, and 
have incircles centered
at the corresponding vertices of its incentral
triangle (F. M. Jackson, pers. comm., Feb. 13, 2010). For an obtuse triangle, two of these quadrilaterals
are concave polygons, and the associated circles
touch extensions of some of the sides.
Let 
,
,
and 
be the side lengths of the reference
triangle, 
,
,
and 
its corresponding angles, and 
its area. The side
lengths of the reflection triangle are
 |  |  |
(2)
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 |  |  |
(3)
|
 |  |  |
(4)
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(Janecke 2011b).
Its area 
is given by
 |  |  |
(5)
|
 |  |  |
(6)
|
(P. Moses, pers. comm., Jan. 31, 2005), where 
is the circumcenter, 
is the orthocenter,
is the circumradius, and 
is the area of the reference
triangle.
The area can also be expressed as
 |
(7)
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(Janecke 2011b).
If the reference triangle is an acute triangle, then its reflection triangle is also an acute
triangle. Writing 
, 
, and 
for the angles of the reflection
triangle gives the inequality
 |
(8)
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(Janecke 2011a), with equality iff the reference triangle is an equilateral triangle.
Its triangle centroid is Kimberling center 
(Kimberling), with triangle center function
 |  |  |
(9)
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 |  |  |
(10)
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(P. Moses, pers. comm., Feb. 7, 2005), where 
, 
, 
, and 
are Conway triangle
notation. The circumcircle of the reflection
triangle is the reflection circle, and its circumcenter is Kimberling
center 
,
which is the 
-Ceva conjugate of 
. Its orthocenter is Kimberling
center 
(Kimberling).
The reflection triangle is perspective to the Cevian triangles with Cevian
points lying on the orthopivotal cubic curve K060,
including Kimberling centers 
for 
, 5, 13, 14, 30, 79, 80, 265, 621, 622, 1117, and 1141 (Gibert).
It is perspective to the anticevian
triangles with anticevian points lying on the Napoleon-Feuerbach
cubic, corresponding to Kimberling centers
with 
,
3, 4, 5, 17, 18, 54, 61, 62, 195, 627, 628, 2120, and 2121. It is also perspective
to the antipedal triangles with antipedal points corresponding to Kimberling
centers with 
,
5, 20, 24, 54, 64, 68, 155, 254, and 2917 (P. Moses, pers. comm., Feb. 3,
2005).
The reflection triangle is degenerate iff
 |
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(Bottema 1987).
The reflection triangle is homothetic to the pedal triangle of the nine-point center (Bottema
1987). In particular, if 
is the triangle centroid
of 
,
then the reflection triangle is the image of the pedal
triangle of the nine-point center under
the homothecy 
(Boutte 2001, cited in Grinberg 2003).
The circumcenter of the reflection triangle is Kimberling center 
, which is the 
-Ceva conjugate of 
.
Up to similarity, exactly three nondegenerate triangles are similar to their reflection triangles. They are the
equilateral triangle, the heptagonal
triangle, and the obtuse triangle with angles 
, 
, and 
defined by
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(12)
|
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(13)
|
(Nicollier 2012a). The equilateral triangle and heptagonal triangle are directly
similar to their reflection triangles, while the obtuse
triangle above is inversely similar to its
reflection triangle (Nicollier 2012a). The angles 
and 
are the inverse tangents
of algebraic numbers of degree
24 over the rationals.
For an acute triangle or a right triangle, repeated iteration of the reflection
construction makes all three angles tend to 
, so the limit shape is an equilateral triangle (Nicollier 2012a).
