Routh's theorem

The set of Routh , named after Edward Routh , is a mathematical theorem to the geometry of the triangle . He makes the following statement about the area of triangles (see graphic):

Let ABC be a triangle with area (outer triangle in the graphic). Furthermore, let F , D and E points on the sides [ AB ], [ BC ] and [ AC ] respectively . The partial ratios are: ${\ displaystyle A_ {ABC}}$

{\ displaystyle {\ overline {AF}} / {\ overline {BF}} = r}

{\ displaystyle {\ overline {BD}} / {\ overline {CD}} = s}

{\ displaystyle {\ overline {CE}} / {\ overline {AE}} = t}

The intersections of AD and CF , AD and BE or BE and CF are designated by I , G and H.

Then applies to the area of triangle GHI (inner triangle in the graphic):

{\ displaystyle A_ {GHI} = {\ frac {(rst-1) ^ {2}} {(st + s + 1) (rt + t + 1) (rs + r + 1)}} A_ {ABC} .}

The Ceva's Theorem can be considered as a special case of the theorem of Routh. The transversals intersect , and at one point the area of the triangle is equal to 0. From this it can be deduced, i.e. the statement of Ceva's theorem. ${\ displaystyle [AD]}$ ${\ displaystyle [BE]}$ ${\ displaystyle [CF]}$ ${\ displaystyle GHI}$ ${\ displaystyle rst = 1}$

literature

HSM Coxeter : Introduction to Geometry. 2nd Edition. Wiley, New York NY et al. a. 1969, ISBN 0-471-18283-4 , pp. 211, 219-220.
Ivan Niven : A New Proof of Routh's Theorem . In: Mathematics Magazine , Volume 49, No. 1 (Jan., 1976), pp. 25-27 ( JSTOR 2689876 )

Web links

Eric W. Weisstein : Routh's Theorem . In: MathWorld (English).
Routh's Formula by Cross Products - Derivation via vector and matrix calculation