Bisector theorem

{\frac {BD}{CD}}={\frac {AB}{AC}}.

{\ displaystyle {\ frac {BD} {CD}} = {\ frac {AB} {AC}}.}

{\ displaystyle {\ frac {BD} {CD}} = {\ frac {AB} {AC}}.}

The bisector theorem is a classical triangle geometry theorem.

Content

1 Formulation
- 1.1 Notes
2 History
3 Variations and generalizations
4 See also
5 notes
6 Literature

Wording

The bisector at the apex of the triangle divides the opposite side into parts proportional to the adjacent sides. That is, if the bisector at the top $A$ ${\ displaystyle A}$ the triangle $\triangle ABC$ ${\ displaystyle \ triangle ABC}$ crosses the side $B C$ ${\ displaystyle BC}$ at the point $D$ ${\ displaystyle D}$ then

{\frac {DB}{DC}}={\frac {AB}{AC}}.

{\ displaystyle {\ frac {DB} {DC}} = {\ frac {AB} {AC}}.}

Remarks

The same equality holds for the point $D$ ${\ displaystyle D}$ lying at the intersection of the external bisector and the continuation of the side $B C$ ${\ displaystyle BC}$ .

History

The bisectrix theorem is formulated in the sixth book “The Beginning of Euclid ” (Proposition III) ^[1] , in particular, in Greek in the Byzantine manuscript ^[2] . An early Euclidean quote of this theorem in Russian-language sources is contained in one of the first Russian geometry textbooks - the manuscript of the beginning of the 17th century “ Synodal No. 42 ” (book 1, part 2, chapter 21).

Variations and generalizations

If D is an arbitrary point on the side BC of triangle ABC , then
${\frac {|BD|}{|DC|}}={\frac {|AB|{\frac {\sin \angle DAB}{\sin \angle ADB}}}{|AC|{\frac {\sin \angle DAC}{\sin \angle ADC}}}}={\frac {|AB|{\frac {\sin \angle DAB}{\sin \angle ADB}}}{|AC|{\frac {\sin \angle DAC}{\sin(180^{\circ }-\angle ADB))}}}}={\frac {|AB|\sin \angle DAB}{|AC|\sin \angle DAC}}.$ ${\ displaystyle {\ frac {| BD |} {| DC |}} = {\ frac {| AB | {\ frac {\ sin \ angle DAB} {\ sin \ angle ADB}}} {| AC | {\ frac {\ sin \ angle DAC} {\ sin \ angle ADC}}}} = {\ frac {| AB | {\ frac {\ sin \ angle DAB} {\ sin \ angle ADB}}} {| AC | { \ frac {\ sin \ angle DAC} {\ sin (180 ^ {\ circ} - \ angle ADB))}}}} = {\ frac {| AB | \ sin \ angle DAB} {| AC | \ sin \ angle DAC}}.}$
- In the case when AD is a bisector, $\sin \angle DAB=\sin \angle DAC$ ${\ displaystyle \ sin \ angle DAB = \ sin \ angle DAC}$ .

The bisector plane of the dihedral angle in the tetrahedron (that is, the plane dividing the dihedral angle in half) divides its opposite edge into parts proportional to the areas of the faces of the tetrahedron that are the faces of this dihedral angle ^[3] .

Steiner theorem .

Notes

↑ The Euclidean began eight books, namely: the first six, 11th and 12th, containing the foundations of geometry. / Per. F. Petrushevsky. - SPb. , 1819. - S. 205. - 480 p.
↑ Bisectrix theorem in the Byzantine manuscript
↑ Gusyatnikov P.B., Reznichenko S.V. Vector algebra in examples and problems . - M .: Higher school , 1985 .-- 232 p.

Literature

Ponarin I.P. Elementary geometry. In 2 vols. - M .: ICMMO , 2004 .-- S. 18-19.

[1] The Euclidean began eight books, namely: the first six, 11th and 12th, containing the foundations of geometry. / Per. F. Petrushevsky. - SPb. , 1819. - S. 205. - 480 p.

[2] Bisectrix theorem in the Byzantine manuscript

[3] Gusyatnikov P.B., Reznichenko S.V. Vector algebra in examples and problems . - M .: Higher school , 1985 .-- 232 p.