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Reflection Triangle

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ReflectedTriangle

The triangle DeltaA^*B^*C^* obtained by reflecting the vertices of a reference triangle DeltaABC about the opposite sides is called the reflection triangle (Grinberg 2003). It is perspective to the reference triangle with the orthocenter H as the perspector, and has trilinear vertex matrix

[-1 2cosC 2cosB; 2cosC -1 2cosA; 2cosB 2cosA -1].
(1)

Its side lengths are

a^'=1/(bc)(sqrt(a^6-3b^2a^4-3c^2a^4+3b^4a^2+3c^4a^2+3b^2c^2a^2-b^6-c^6+b^2c^4+b^4c^2))
(2)
b^'=1/(ac)(sqrt(-a^6+3b^2a^4+c^2a^4-3b^4a^2+c^4a^2+3b^2c^2a^2+b^6-c^6+3b^2c^4-3b^4c^2))
(3)
c^'=1/(ab)(sqrt(-a^6+b^2a^4+3c^2a^4+b^4a^2-3c^4a^2+3b^2c^2a^2-b^6+c^6-3b^2c^4+3b^4c^2)).
(4)

Its area is given by

Delta=4-((OH)/R)^2
(5)
=-(a^6-b^2a^4-c^2a^4-b^4a^2-c^4a^2-b^2c^2a^2+b^6+c^6-b^2c^4-b^4c^2)/(a^2b^2c^2)Delta
(6)

(P. Moses, pers. comm., Jan. 31, 2005), where O is the circumcenter, H is the orthocenter, R is the circumradius, and Delta is the area of the reference triangle.

Its triangle centroid has triangle center function

alpha=a(S^2-S_A^2+2S_BS_C)
(7)
=a(-b^4+a^2b^2+c^2b^2-c^4+a^2c^2),
(8)

which is not a Kimberling center (P. Moses, pers. comm., Feb. 7, 2005), where S, S_A, S_B, and S_C are Conway triangle notation. The circumcircle of the reflection triangle is the reflection circle, and its circumcenter is Kimberling center X_(195), which is the X_5-Ceva conjugate of X_3. Its orthocenter has a complicated triangle center function that is not a Kimberling center.

The reflection triangle is perspective to the Cevian triangles with Cevian points lying on the orthopivotal cubic K060, corresponding to Kimberling centers X_i for i=4, 5, 13, 14, 30, 79, 80, 621, 622, 1117, and 1141. It is perspective to the anticevian triangles with anticevian points lying on the Napoleon-Feuerbach cubic, corresponding to Kimberling centers with i=1, 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 i=1, 5, 20, 24, 54, 64, 68, 155, 254, and 2917 (P. Moses, pers. comm., Feb. 3, 2005).

The reflection triangle is degenerate iff

cosAcosBcosC=-3/8
(9)

(Bottema 1987).

The reflection triangle is homothetic to the pedal triangle of the nine-point circle (Bottema 1987). In particular, if G is the triangle centroid of DeltaABC, then the reflection triangle is the image of the pedal triangle of the nine-point center under the homothecy h(G,4) (Boutte 2001, cited in Grinberg 2003).

The circumcenter of the reflection triangle is Kimberling center X_(195), which is the X_5-Ceva conjugate of X_3.

See also

Reflection Circle, Reflection

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References

Bottema, O. Hoofdstukken uit de elementaire meetkunde, 2nd ed. Utrecht, Netherlands: Epsilon, pp. 83-87, 1987.Grinberg, D. "On the Kosnita Point and the Reflection Triangle." Forum Geom. 3, 105-111, 2003. http://forumgeom.fau.edu/FG2003volume3/FG200311index.html.

Referenced on Wolfram|Alpha

Reflection Triangle

Cite this as:

Weisstein, Eric W. "Reflection Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ReflectionTriangle.html

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