Editing Pedal triangle (section)

{{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>