Bisector
Contents
[hide]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 A
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 line
meet
at point
and
at point
Denote circumcenter of
- the point where bisector
meet circumcircle of
Prove:
c) lines and
are concurrent at
Proof
WLOG, A few preliminary formulas:
a)
b)
is the circumcenter of
c) are collinear.
are collinear and so on. Using Cheva's theorem we get the result.
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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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Bisectrix and bisector
Let triangle be given.
Let be the incenter,
be the circumcenter,
be the circumcircle of
Let
be the bisector of
Let
be the circumcenter of
Prove that the points and
are concyclic and
Proof
Denote the midpoint
It is known that is the midpoint
(see Bisector and circumcircle)
the homothety centered at point
with ratio
maps
into
and
into
Point is symmetrical to point
with respect to the line
so radii of
and
are equal.
Denote the radius of
the radius of
Then
Similarly,
So
Therefore, the homothety centered at point with ratio
maps quadrangle
into quadrangle
and
into
points
and
are concyclic.
Similarly, points and
are concyclic. So points
and
are concyclic.
The center of this circle lyes on radius
of
and
The homothety centered at point with ratio
maps the point
(midpoint
) into midpoint
so
Corollary
Points and
are concyclic
Points and
are concyclic.
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Seven lines crossing point
Let be the incenter, circumcircle, and the midpoints of sides
of a
Let be the bisectors of a
be the midpoint of
The points and
be such points that
Denote points
Prove that the lines and the tangent to the circumcircle of
at
are concurrent.
Proof
1. Denote
Similarly
is the bisector of
Similarly,
is the bisector of
is the bisector of
Therefore are rhombus.
So triples of points are collinear, lines
It is known that
Similarly,
is the bisector
Similarly,
Denote the crosspoint of the tangent to the circumcircle of
at
and
is the bisector
2. Let us consider the points and
We use Menelaus' Theorem for and line
and get
We use Menelaus' Theorem for
and get that points
and
are collinear.
3.Let us consider the points and
We use Menelaus' Theorem for
and line
and get
Let
be the midpoint
So
We use Menelaus' Theorem for and get that points
and
are collinear.
Similarly points and
are collinear.
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