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Chisinau MO p139 1977 VIII angle bisector, \beta^2=ac- b^2ac/(a+c)^2

Source:

March 17, 2021

geometryangle bisector

Problem Statement

Let

\beta

be the length of the bisector of angle

B

, and

a', c'

be the lengths of the segments into which this bisector divides the side

AC

of the triangle

ABC

. Prove the relation

\beta^2 = ac-a'c'

and derive from this the formula

\beta^2=ac-\frac{b^2ac}{(a+c)^2}