PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (51)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Copyright
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] trigonometric version of Ceva's theorem (Theorem)

Let $ABC$ be a given triangle and $P$ any point of the plane. If $X$ is the intersection point of $AP$ with $BC$, $Y$ the intersection point of $BP$ with $CA$ and $Z$ is the intersection point of $CP$ with $AB$, then

\begin{displaymath}\frac{\sin ACZ}{\sin ZCB}\cdot\frac{\sin BAX}{\sin XAC}\cdot\frac{\sin CBY}{\sin YBA}=1.\end{displaymath}

Conversely, if $X,Y,Z$ are points on $BC,CA,AB$ respectively, and if

\begin{displaymath} \frac{\sin ACZ}{\sin ZCB}\cdot\frac{\sin BAX}{\sin XAC}\cdot\frac{\sin CBY}{\sin YBA}=1 \end{displaymath}

then $AD,BE,CF$ are concurrent.

Remarks: All the angles are directed angles (counterclockwise is positive), and the intersection points may lie in the prolongation of the segments.

\includegraphics[scale=0.75]{ceva}



"trigonometric version of Ceva's theorem" is owned by drini. [ owner history (1) ]
(view preamble | get metadata)

View style:

See Also: triangle, median, centroid, orthocenter, orthic triangle, cevian, incenter, Gergonne point, Menelaus' theorem, proof of Van Aubel theorem, Van Aubel theorem

Keywords:  Concurrence, Menelaus Theorem

This object's parent.

Attachments:
proof of trigonometric version of Ceva's theorem (Proof) by drini
Log in to rate this entry.
(view current ratings)

Cross-references: segments, positive, angles, concurrent, conversely, intersection, plane, point, triangle

This is version 3 of trigonometric version of Ceva's theorem, born on 2004-11-17, modified 2005-01-19.
Object id is 6483, canonical name is TrigonometricVersionOfCevasTheorem.
Accessed 2032 times total.

Classification:
AMS MSC: 51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)