Editing Pedal triangle

{{short description|Triangle found by projecting a point onto the sides of another triangle}}
[[File:Pedal Triangle.svg|right|thumb|
{{legend-line|solid black|Triangle {{math|△''ABC''}}}}
{{legend-line|solid #0373fc|[[Perpendiculars]] from point {{mvar|P}}}}
{{legend-line|solid red|Obtained pedal triangle {{math|△''LMN''}}}}
]]

In [[plane geometry]], a '''pedal triangle''' is obtained by projecting a  [[point (geometry)|point]] onto the sides of a [[triangle]].

More specifically, consider a triangle {{math|△''ABC''}}, and a point {{mvar|P}} that is not one of the vertices {{mvar|A, B, C}}. Drop [[perpendiculars]] from {{mvar|P}} to the three sides of the triangle (these may need to be produced, i.e., [[extended side|extended]]).  Label {{mvar|L, M, N}} the [[Line–line intersection|intersections]] of the lines from {{mvar|P}} with the sides {{mvar|BC, AC, AB}}. The pedal triangle is then {{math|△''LMN''}}.

If {{math|△''ABC''}} is not an [[obtuse triangle]] and {{mvar|P}} is the [[orthocenter]], then the angles of {{math|△''LMN''}} are {{math|180° − 2''A''}}, {{math|180° − 2{{mvar|B}}}} and {{math|180° − 2''C''}}.<ref>{{Cite web|title=Trigonometry/Circles and Triangles/The Pedal Triangle - Wikibooks, open books for an open world|url=https://en.wikibooks.org/wiki/Trigonometry/Circles_and_Triangles/The_Pedal_Triangle#:~:text=As%20already%20noted,%20the%20altitudes,ABC%20is%20its%20excentral%20triangle.&text=If%20ABC%20is%20not%20an,and%20its%20sides%20are%20a.|access-date=2020-10-31|website=en.wikibooks.org}}</ref>

The quadrilaterals {{mvar|PMAN, PLBN, PLCM}} are [[cyclic quadrilaterals]].

The location of the chosen point {{mvar|P}} relative to the chosen triangle {{math|△''ABC''}} gives rise to some special cases:

* If {{mvar|P}} is the [[orthocenter]], then {{math|△''LMN''}} is the [[orthic triangle]].
* If {{mvar|P}} is the [[incenter]], then {{math|△''LMN''}} is the [[intouch triangle]].
* If {{mvar|P}} is the [[circumcenter]], then {{math|△''LMN''}} is the [[medial triangle]].
*If {{mvar|P}} is on the [[circumcircle]] of the triangle, {{math|△''LMN''}} collapses to a line (the ''pedal line'' or ''[[Simson line]]'').

[[File:Pedal Line.svg|right|thumb|'''Special case:''' {{mvar|P}} is on the [[circumcircle]].
{{legend-line|solid black|Triangle {{math|△''ABC''}}}}
{{legend-line|solid #66cc66|Circumcircle of {{math|△''ABC''}}}}
{{legend-line|solid #0373fc|Perpendiculars from {{mvar|P}}}}
{{legend-line|solid red|Obtained pedal line {{mvar|LMN}}}}
]]

The vertices of the pedal triangle of an interior point {{mvar|P}}, as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy [[Carnot's theorem (perpendiculars)|Carnot's theorem]]:<ref>{{Cite book|title=Challenging problems in geometry|url=https://archive.org/details/challengingprobl00posa|url-access=limited|author1=Alfred S. Posamentier|author-link=Alfred S. Posamentier|author2=Charles T. Salkind|isbn=9780486134864|location=New York|oclc=829151719|publisher=Dover|year=1996|pages=[https://archive.org/details/challengingprobl00posa/page/n95 85]-86}}</ref>

<math display=block>|AN|^2 + |BL|^2 + |CM|^2 = |NB|^2 + |LC|^2 + |MA|^2.</math>

==Trilinear coordinates==
If {{mvar|P}} has [[trilinear coordinates]] {{math|''p'' : ''q'' : ''r''}}, then the vertices {{mvar|L, M, N}} of the pedal triangle of {{mvar|P}} are given by
<math display=block>\begin{array}{ccccccc}
  L &=& 0 &:& q+p\cos C &:& r+p\cos B \\[2pt]
  M &=& p+q\cos C &:& 0 &:& r+q\cos A \\[2pt]
  N &=& p+r\cos B &:& q+r\cos A &:& 0
\end{array}</math>

==Antipedal triangle==

One vertex, {{mvar|L'}}, of the '''antipedal triangle''' of {{mvar|P}} is the point of intersection of the perpendicular to {{mvar|BP}} through {{mvar|B}} and the perpendicular to {{mvar|CP}} through {{mvar|C}}. Its other vertices, {{mvar|M'}} and {{mvar|N'}}, are constructed analogously.  [[Trilinear coordinates]] are given by
<math display=block>\begin{array}{ccrcrcr}
  L' &=& -(q+p\cos C)(r+p\cos B) &:& (r+p\cos B)(p+q\cos C) &:& (q+p\cos C)(p+r\cos B) \\[2pt]
  M' &=& (r+q\cos A)(q+p\cos C) &:& -(r+q\cos A)(p+q\cos C) &:& (p+q\cos C)(q+r\cos A) \\[2pt]
  N' &=& (q+r\cos A)(r+p\cos B) &:& (p+r\cos B)(r+q\cos A) &:& -(p+r\cos B)(q+r\cos A)
\end{array}</math>


For example, the [[excentral triangle]] is the antipedal triangle of the incenter.

Suppose that {{mvar|P}} does not lie on any of the extended sides {{mvar|BC, CA, AB}}, and let {{math|''P''<sup> −1</sup>}} denote the [[isogonal conjugate]] of {{mvar|P}}.  The pedal triangle of {{mvar|P}} is [[Homothetic transformation|homothetic]] to the antipedal triangle of {{math|''P''<sup> −1</sup>}}.  The homothetic center (which is a triangle center if and only if {{mvar|P}} is a triangle center) is the point given in [[trilinear coordinates]] by

<math display=block>ap(p+q\cos C)(p+r\cos B) \ :\ bq(q+r\cos A)(q+p\cos C) \ :\ cr(r+p\cos B)(r+q\cos A)</math>

The product of the areas of the pedal triangle of {{mvar|P}} and the antipedal triangle of {{math|''P''<sup> −1</sup>}} equals the square of the area of {{math|△''ABC''}}.

== Pedal circle ==
{{Main|Pedal circle}}
[[File:Isogonal points2 pedal circle.svg|thumb|336x336px|The pedal circle of the point {{mvar|P}} and its isogonal conjugate {{mvar|P*}} are the same.]]
The pedal circle is defined as the [[circumcircle]] of the pedal triangle. Note that the pedal circle is not defined for points lying on the circumcircle of the triangle. 

=== Pedal circle of isogonal conjugates ===
For any point {{mvar|P}} not lying on the circumcircle of the triangle, it is known that {{mvar|P}} and its [[isogonal conjugate]] {{mvar|P*}} have a common pedal circle, whose center is the midpoint of these two points.<ref>{{Cite book|last=Honsberger|first=Ross|url=http://dx.doi.org/10.5948/upo9780883859513|title=Episodes in Nineteenth and Twentieth Century Euclidean Geometry|date=1995-01-01|publisher=The Mathematical Association of America|isbn=978-0-88385-951-3}}</ref>

==References==

{{Reflist}}

== External links ==
* [http://mathworld.wolfram.com/PedalTriangle.html Mathworld: Pedal Triangle] 
* [http://www.cut-the-knot.org/ctk/SimsonLine.shtml#pedalTriangle Simson Line]
* [http://www.cut-the-knot.org/Curriculum/Geometry/OrthologicPedal.shtml Pedal Triangle and Isogonal Conjugacy]
* [https://www.geogebra.org/m/strsmxt5 pedal triangle and pedal circle] - interactive illustration
[[Category:Objects defined for a triangle]]