Van Aubel's Triangle Theorem

The Van Obel triangle theorem is a classical affine geometry theorem.

Wording

The case when all three points lie on the sides of the triangle, and not on their extensions.

The case when two points lie on the extensions of the sides.

If direct $A P$ ${\ displaystyle AP}$ , $B P$ ${\ displaystyle BP}$ , $C P$ ${\ displaystyle CP}$ straight lines cross respectively $B C$ ${\ displaystyle BC}$ , $C A$ ${\ displaystyle CA}$ , $A B$ ${\ displaystyle AB}$ containing sides of a triangle $A B C$ ${\ displaystyle ABC}$ , respectively, at points $A_{1}$ ${\ displaystyle A_ {1}}$ , $B_{1}$ ${\ displaystyle B_ {1}}$ , $C_{1}$ ${\ displaystyle C_ {1}}$ , then there is an equality of relations of directed segments :

{\frac {AC_{1}}{C_{1}B}}+{\frac {AB_{1}}{B_{1}C}}={\frac {AP}{PA_{1}}}

{\ displaystyle {\ frac {AC_ {1}} {C_ {1} B}} + {\ frac {AB_ {1}} {B_ {1} C}} = {\ frac {AP} {PA_ {1} }}}

.

Remarks

If the segments are co-directional (equally directed), then the upper signs of the directed segments can be removed, and we obtain a scalar version of the Van Aubel theorem:
${\frac {AC_{1}}{C_{1}B}}+{\frac {AB_{1}}{B_{1}C}}={\frac {AP}{PA_{1}}}$ ${\ displaystyle {\ frac {AC_ {1}} {C_ {1} B}} + {\ frac {AB_ {1}} {B_ {1} C}} = {\ frac {AP} {PA_ {1} }}}$ .

Van Aubel's Triangle Theorem

Wording

Remarks

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