Routh's Theorem gives a formula for the area of the triangle formed by three cevians, one drawn from each vertex of a given triangle to a point on the opposite side dividing it in a specified ratio, expressing that inner triangle's area as an explicit fraction of the original triangle's area. Named for Edward John Routh, it generalizes the simpler case known as Ceva's Theorem, which addresses only when the three cevians are concurrent.
Facts
StatementThe signed area of the triangle formed by the cevians AD, BE and CF, expressed in terms of the ratios x, y and z in which each cevian divides its side, equals the area of triangle ABC times the quantity xyz minus 1, squared, divided by the product of xy plus y plus 1, yz plus z plus 1, and zx plus x plus 1. 1 Classification
Statement Form Connections
Has Statement Form
In Branch
Source Routh's theorem - Wikipedia
Proved By
Source Routh's theorem - Wikipedia
Sources
1. Routh's theorem - Wikipedia
Proof section
the signed area of the triangle formed by the cevians AD, BE, and CF is S_ABC times (xyz - 1) squared divided by (xy + y + 1)(yz + z + 1)(zx + x + 1)
Citations section
Routh stated the theorem already in the first edition of 1891, Volume 1, Chap. IV
In Branch: Geometry, Lead sentence
In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersec
Proved By: Edward John Routh, Lead sentence, attribution
This theorem was given by Edward John Routh
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