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Problems
Contests
National and Regional Contests
Moldova Contests
Chisinau City MO
1977 Chisinau City MO
139
139
Part of
1977 Chisinau City MO
Problems
(1)
Chisinau MO p139 1977 VIII angle bisector, \beta^2=ac- b^2ac/(a+c)^2
Source:
3/17/2021
Let
β
\beta
β
be the length of the bisector of angle
B
B
B
, and
a
′
,
c
′
a', c'
a
′
,
c
′
be the lengths of the segments into which this bisector divides the side
A
C
AC
A
C
of the triangle
A
B
C
ABC
A
BC
. Prove the relation
β
2
=
a
c
−
a
′
c
′
\beta^2 = ac-a'c'
β
2
=
a
c
−
a
′
c
′
and derive from this the formula
β
2
=
a
c
−
b
2
a
c
(
a
+
c
)
2
\beta^2=ac-\frac{b^2ac}{(a+c)^2}
β
2
=
a
c
−
(
a
+
c
)
2
b
2
a
c
.
geometry
angle bisector