Editing Orthocenter (section)

==Orthic triangle==<!-- This section is linked from [[Fagnano problem]] -->
[[File:Altitudes and orthic triangle SVG.svg|thumb|Triangle {{math|△''abc''}} (respectively, {{math|△''DEF''}} in the text) is the orthic triangle of triangle {{math|△''ABC''}}]]
If the triangle {{math|△''ABC''}} is [[Acute and obtuse triangles|oblique]] (does not contain a right-angle), the [[pedal triangle]] of the orthocenter of the original triangle is called the '''orthic triangle''' or '''altitude triangle'''. That is, the feet of the altitudes of an oblique triangle form the orthic triangle, {{math|△''DEF''}}. Also, the incenter (the center of the inscribed circle) of the orthic triangle {{math|△''DEF''}} is the orthocenter of the original triangle {{math|△''ABC''}}.<ref name=Barker>
{{cite book |title=Continuous symmetry: from Euclid to Klein |author= William H. Barker, Roger Howe |chapter-url=https://books.google.com/books?id=NIxExnr2EjYC&pg=PA292 |chapter=§ VI.2: The classical coincidences |isbn=978-0-8218-3900-3 |publisher=American Mathematical Society|year=2007|page= 292}} See also: Corollary 5.5, p. 318.
</ref>

[[Trilinear coordinates]] for the vertices of the orthic triangle are given by
<math display=block>\begin{array}{rccccc}
D =& 0 &:& \sec B &:& \sec C \\
E =& \sec A &:& 0 &:& \sec C \\
F =& \sec A &:& \sec B &:& 0
\end{array}</math>

The [[extended side]]s of the orthic triangle meet the opposite extended sides of its reference triangle at three [[collinear points]].<ref>{{harvnb|Johnson|2007|loc=p. 199, Section 315}}</ref><ref>{{harvnb|Altshiller-Court|2007|loc=p. 165}}</ref><ref name=Barker />

In any [[acute triangle]], the inscribed triangle with the smallest perimeter is the orthic triangle.<ref name=Johnson>{{harvnb|Johnson|2007|loc=p. 168, Section 264}}</ref> This is the solution to [[Fagnano's problem]], posed in 1775.<ref>{{harvnb|Berele|Goldman|2001|loc=pp. 120-122}}</ref> The sides of the orthic triangle are parallel to the tangents to the circumcircle at the original triangle's vertices.<ref>{{harvnb|Johnson|2007|loc = p. 172, Section 270c}}</ref>

The orthic triangle of an acute triangle gives a triangular light route.<ref>Bryant, V., and Bradley, H., "Triangular Light Routes," ''Mathematical Gazette'' 82, July 1998, 298-299.</ref>

The tangent lines of the nine-point circle at the midpoints of the sides of {{math|△''ABC''}} are parallel to the sides of the orthic triangle, forming a triangle similar to the orthic triangle.<ref>{{citation|first=David C.|last=Kay|title=College Geometry / A Discovery Approach|year=1993|publisher=HarperCollins|isbn=0-06-500006-4|page=6}}</ref>

The orthic triangle is closely related to the [[tangential triangle]], constructed as follows:  let {{mvar|L{{sub|A}}}} be the line tangent to the circumcircle of triangle {{math|△''ABC''}} at vertex {{mvar|A}}, and define {{mvar|L{{sub|B}}, L{{sub|C}}}} analogously.  Let <math>A'' = L_B \cap L_C,</math> <math>B'' = L_C \cap L_A,</math> <math>C'' = L_C \cap L_A.</math>  The tangential triangle is  {{math|△''A"B"C"''}}, whose sides are the tangents to triangle {{math|△''ABC''}}'s circumcircle at its vertices; it is [[Homothetic transformation|homothetic]] to the orthic triangle. The circumcenter of the tangential triangle, and the [[center of similitude]] of the orthic and tangential triangles, are on the [[Euler line]].<ref name="SL"/>{{rp|p. 447}}

Trilinear coordinates for the vertices of the tangential triangle are given by
<math display=block>\begin{array}{rrcrcr}
A'' =& -a &:& b &:& c \\
B'' =& a &:& -b &:& c \\
C'' =& a &:& b &:& -c
\end{array}</math>
The reference triangle and its orthic triangle are [[orthologic triangles]].

For more information on the orthic triangle, see [[Orthocentric system#The common orthic triangle, its incenter and excenters|here]].