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 vladimir.shelomovskii@gmail.com, vvsss
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) vladimir.shelomovskii@gmail.com, vvsss
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 vladimir.shelomovskii@gmail.com, vvsss
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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