# Symmetry

A proof utilizes **symmetry** if the steps to prove one thing is identical to those steps of another. For example, to prove that in triangle ABC with all three sides congruent to each other that all three angles are equal, you only need to prove that if then the other cases hold by symmetry because the steps are the same.

## Contents

- 1 Hidden symmetry
- 2 Symmetry with respect angle bisectors
- 3 Symmetry with respect angle bisectors 1
- 4 Construction of triangle
- 5 Symmetry with respect angle bisectors 2
- 6 Symmetry of radical axes
- 7 Composition of symmetries
- 8 Composition of symmetries 1
- 9 Composition of symmetries 2
- 10 Symmetry and secant
- 11 Symmetry and incircle
- 12 Symmetry and incircle A
- 13 Symmetry for 60 degrees angle
- 14 See also

## Hidden symmetry

Let the convex quadrilateral be given.

Prove that

**Proof**

Let be bisector

Let point be symmetric with respect

is isosceles.

Therefore
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## Symmetry with respect angle bisectors

Given the triangle is the incircle, is the incenter,

Points and are symmetrical to point with respect to the lines containing the bisectors and respectively.

Prove that is the midpoint

* Proof*
Denote

The tangents from point to are equal

Point is symmetrical to point with respect is symmetrical to segment

Symilarly,
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## Symmetry with respect angle bisectors 1

The bisector intersect the incircle of the triangle at the point The point is symmetric to with respect to the point is symmetric to with respect to Prove that is the bisector of the segment

**Proof**

The point is symmetric to with respect to

The point is symmetric to with respect to

Similarly

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## Construction of triangle

Given points and at which the segments of the bisectors and respectively intersect the incircle of centered at

Construct the triangle

**Construction**

We construct the incenter of as circumcenter of

If these points are collinear or if construction is impossible.

We construct bisectors and

We construct the points and symmetrical to point with respect to and respectively.

We construct the bisector and choose the point as the point intersection with the circle closest to the line

We construct a tangent to the the circle at the point It intersects the lines and at points and respectively.

We construct the tangents to which are symmetrical to sideline with respect to and

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## Symmetry with respect angle bisectors 2

Given the triangle is the incircle, is the incenter,

Let be the point on sideline

Points and are symmetrical to point with respect to the lines and respectively. The line contains point

Prove that is the midpoint

**Proof**

The segment is symmetric to with respect to the segment is symmetric to with respect to So

Similarly at midpoint

or
or
We use the Law of Sines and get:
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## Symmetry of radical axes

Let triangle be given. The point and the circle are the incenter and the circumcircle of

Circle centered at has the radius and intersects at points and Line is the tangent for at the point

Prove that line is symmetry to the line with respect axis

**Proof**

circle centered at contain points and and is tangent for and

is the radical axis of and is the radical axis of and

is the radical axis of and and are concurrent (at point )

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## Composition of symmetries

Let the inscribed convex hexagon be given, Prove that

**Proof**

Denote the circumcenter of

the common bisector the common bisector

the smaller angle between lines and

is the symmetry with respect axis is the symmetry with respect axis

It is known that the composition of two axial symmetries with non-parallel axes is a rotation centered at point of intersection of the axes at twice the angle from the axis of the first symmetry to the axis of the second symmetry.

Therefore
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## Composition of symmetries 1

Let the triangle be given.

is the incircle, is the incenter, is the circumcenter of The point is symmetric to with respect to is symmetric to with respect to is symmetric to with respect to

Prove: a)

b)

**Proof**

a) Denote the smaller angle between and

is the symmetry with respect axis is the symmetry with respect axis

counterclockwise direction.

clockwise direction.

Therefore is parallel to tangent line for at point

b) is homothetic to

is the circumcenter of

The center of the homothety lies on the line passing through the circumcenters of the triangles.

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## Composition of symmetries 2

Let triangle be given. The point and the circle are the incenter and the incircle of

Let be the symmetry with respect axis be the symmetry with respect axis the symmetry with respect axis Find the composition of axial symmetries with respect and

**Solution**

It is known that the composition of three axial symmetries whose axes intersect at one point is an axial symmetry whose axis contains the same point

Consider the composition of axial symmetries for point

is a fixed point of transformation.

This means that the desired axis of symmetry contains points and , this is a straight line

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## Symmetry and secant

The circle centered at and the point be given. Let and be the tangents, be the secant (

Segment intersects segment at point Prove that

**Proof**

Let be symmetric to with respect the line

It is known that We use symmetry and get It is known that

Triangles and have common side

Similar triangles and have the areas ratio

Therefore

According the Cross-ratio criterion the four points are a harmonic range (on the real projective line).

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## Symmetry and incircle

Let with incircle be given.

Point

1. Let be the point in

Denote

Prove that

2. Let be the point in the segment

Let be the point in the ray such that

Denote

Prove that points and are collinear.

**Proof**

We use Menelaus theorem for a triangle and a transversal line and get:

2. Denote Then
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## Symmetry and incircle A

Denote is the arbitrary point.

Prove that and are concurrent.

**Proof**

Denote is the incenter of

Let us make the projective transformations mapping circle onto circle and point onto center of this circle.

Denote the result of transformation of point This transformation maps point to infinity.

Segment this transformation maps onto diameter onto

We use the cross-ratio which is fixed, equation the Claim, and get

so is the midpoint of

Point in infinity, so

Lines and are crossing at the line of symmetry therefore lines and are concurrent.

**Claim**

Let with incircle be given.

Point Prove

**Proof**

WLOG, Denote the point in such that
Denote
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## Symmetry for 60 degrees angle

Let an isosceles triangle be given.

Let be the bisector of

a) Prove that

b) Prove that

**Proof**

a) One can find successively angles (see diagram).

b) Let

Let

Let

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