Bisector
Contents
Division of bisector
Let a triangle be given.
Let and
be the bisectors of
he segments and
meet at point
Find
Solution
Similarly
Denote
Bisector
Bisector
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Bisectors and tangent
Let a triangle and it’s circumcircle
be given.
Let segments and
be the internal and external bisectors of
The tangent to
at
meet
at point
Prove that
a)
b)
c)
Proof
a)
is circumcenter
b)
c)
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Proportions for bisectors
The bisectors and
of a triangle ABC with
meet at point
Prove
Proof
Denote the angles
and
are concyclic.
The area of the
is
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Bisector and circumcircle
Let a triangle be given.
Let segments
and
be the bisectors of
The lines
and
meet circumcircle
at points
respectively.
Find
Prove that circumcenter
of
lies on
Solution
Incenter
belong the bisector
which is the median of isosceles
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Some properties of the angle bisectors
Let a triangle
be given.
Let be the circumradius, circumcircle, circumcenter, inradius, incircle, and inradius of
respectively.
Let segments and
be the angle bisectors of
lines
and
meet
at
and
meet
and
at
Let be the point on tangent to
at point
such, that
Let bisector
meet
at point
and
at point
Denote circumcenter of
- the point where line
meet circumcircle of
Prove:
c) lines and
are concurrent at
Proof
WLOG, A few preliminary formulas:
- circumcenter
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Proportions for bisectors
The bisectors and
of a triangle ABC with
meet at point
Prove
Proof
Denote the angles
and
are concyclic.
The area of the
is
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