Steiner theorem (planimetry)

Steiner theorem - statement of Euclidean planimetry:

Two vertices are drawn through the vertex A of triangle ABC inside it, forming equal angles with sides AB and AC and intersecting side BC at points M and N. Then ${\frac {BM}{CM}}\cdot {\frac {BN}{CN}}=\left({\frac {AB}{AC}}\right)^{2}$ ${\ displaystyle {\ frac {BM} {CM}} \ cdot {\ frac {BN} {CN}} = \ left ({\ frac {AB} {AC}} \ right) ^ {2}}$ ${\ frac {BM} {CM}} \ cdot {\ frac {BN} {CN}} = \ left ({\ frac {AB} {AC}} \ right) ^ {2}$

An important special case of the theorem

From the Steiner theorem , as a special case, we obtain the bisector theorem . Indeed, suppose that in the theorem stated above the points M and N coincide, forming a point D , then they are the base of the bisector dropped from the vertex A to the side BC . In this particular case, we have ${\frac {BD}{CD}}\cdot {\frac {BD}{CD}}=\left({\frac {AB}{AC}}\right)^{2}$ ${\ displaystyle {\ frac {BD} {CD}} \ cdot {\ frac {BD} {CD}} = \ left ({\ frac {AB} {AC}} \ right) ^ {2}}$ ${\frac {BD}{CD}}\cdot {\frac {BD}{CD}}=\left({\frac {AB}{AC}}\right)^{2}$ . Extracting the square root of both parts, we have ${\frac {BD}{CD}}={\frac {AB}{AC}}$ ${\ displaystyle {\ frac {BD} {CD}} = {\ frac {AB} {AC}}}$ ${\frac {BD}{CD}}={\frac {AB}{AC}}$ , which is the essence of the bisector theorem.

Literature

Ponarin I.P. Elementary geometry. In 2 volumes - M .: ICMNMO , 2004 .-- S. 32. - ISBN 5-94057-170-0 .

Steiner theorem (planimetry)

An important special case of the theorem

Literature

See also

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