Signs of similarity of triangles

Similar triangles in Euclidean geometry are triangles whose angles are respectively equal, and the sides of one triangle are proportional to the similar sides of another triangle.

This article discusses the properties of such triangles in Euclidean geometry. Some statements are incorrect for non-Euclidean geometries.

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Signs of similarity of triangles

The signs of the similarity of triangles are geometric signs that make it possible to establish that two triangles are similar , without using all the elements.

The first sign

If two angles of one triangle are respectively equal to two angles of another triangle, then the triangles are similar.

i.e: $\triangle ABC\sim \triangle A_{1}B_{1}C_{1}\Leftrightarrow \angle A=\angle A_{1},\ \angle B=\angle B_{1}.$ ${\ displaystyle \ triangle ABC \ sim \ triangle A_ {1} B_ {1} C_ {1} \ Leftrightarrow \ angle A = \ angle A_ {1}, \ \ angle B = \ angle B_ {1}.}$

Given: $\triangle ABC$ ${\ displaystyle \ triangle ABC}$ and $\triangle A_{1}B_{1}C_{1},\ \angle A=\angle A_{1},\ \angle B=\angle B_{1}.$ ${\ displaystyle \ triangle A_ {1} B_ {1} C_ {1}, \ \ angle A = \ angle A_ {1}, \ \ angle B = \ angle B_ {1}.}$

Prove: $\triangle ABC\sim \triangle A_{1}B_{1}C_{1}.$ ${\ displaystyle \ triangle ABC \ sim \ triangle A_ {1} B_ {1} C_ {1}.}$

Evidence

From the theorem on the sum of the angles of a triangle it can be obtained that all angles of the triangles are equal. Arrange them so that the angle

\angle BAC

{\ displaystyle \ angle BAC}

overlapped with the corner

\angle B_{1}A_{1}C_{1}

{\ displaystyle \ angle B_ {1} A_ {1} C_ {1}}

. From the generalized Thales theorem (it can be proved without similarity, see for example the textbook on geometry 7-9 by Sharygin or Pogorelov)

AC/A_{1}C_{1}=BC/B_{1}C_{1}

{\ displaystyle AC / A_ {1} C_ {1} = BC / B_ {1} C_ {1}}

. Similarly, it can be proved that the relations are equal and the other respective sides are equal, which means that the triangles are similar by definition, and so on.

Consequences of the First Sign of Similarity

If three different sides of the original triangle are pairwise parallel (twice antiparallel or perpendicular) to three different similar sides of another triangle, then these two triangles are similar . For examples of the application of this corollary, see the sections below: “Examples of similar triangles” and “Properties of parallelism (antiparallelism) of the sides of related triangles”.
Double antiparallel sides mean the following. For example, the sides of a given acute-angled triangle are antiparallel to the corresponding sides of the ortho -triangle against which they lie. In this case, the corresponding sides of the ortho- triangle of the ortho-triangle (twice the ortho- triangle ) are twice antiparallel to the corresponding sides of the original triangle , that is, they are simply parallel. Therefore, the ortho - triangle of the ortho-triangle and the original triangle are similar , for example, as triangles with parallel sides.

Second Sign

If the two sides of one triangle are proportional to the two sides of the other triangle and the angles enclosed between these sides are equal, then such triangles are similar.

Given: $\triangle ABC$ ${\ displaystyle \ triangle ABC}$ and $\triangle A_{1}B_{1}C_{1},\ \angle A=\angle A_{1},\ {\frac {AB}{A_{1}B_{1}}}={\frac {AC}{A_{1}C_{1}}}.$ ${\ displaystyle \ triangle A_ {1} B_ {1} C_ {1}, \ \ angle A = \ angle A_ {1}, \ {\ frac {AB} {A_ {1} B_ {1}}} = { \ frac {AC} {A_ {1} C_ {1}}}.}$

Prove: $\triangle ABC\sim \triangle A_{1}B_{1}C_{1}.$ ${\ displaystyle \ triangle ABC \ sim \ triangle A_ {1} B_ {1} C_ {1}.}$

Evidence

1) Consider $\triangle ABC_{2}$ ${\ displaystyle \ triangle ABC_ {2}}$ , wherein $\angle BAC_{2}=\angle A_{1}$ ${\ displaystyle \ angle BAC_ {2} = \ angle A_ {1}}$ and $\angle ABC_{2}=\angle B_{1}:$ ${\ displaystyle \ angle ABC_ {2} = \ angle B_ {1}:}$

\triangle ABC_{2}\sim \triangle A_{1}B_{1}C_{1}

{\ displaystyle \ triangle ABC_ {2} \ sim \ triangle A_ {1} B_ {1} C_ {1}}

( first sign )

\Rightarrow {\frac {AB}{A_{1}B_{1}}}={\frac {AC_{2}}{A_{1}C_{1}}}.

{\ displaystyle \ Rightarrow {\ frac {AB} {A_ {1} B_ {1}}} = {\ frac {AC_ {2}} {A_ {1} C_ {1}}}.}

2) By condition:

{\frac {AB}{A_{1}B_{1}}}={\frac {AC}{A_{1}C_{1}}}\Rightarrow AC=AC_{2}\Rightarrow \triangle ABC=\triangle ABC_{2}

{\ displaystyle {\ frac {AB} {A_ {1} B_ {1}}} = {\ frac {AC} {A_ {1} C_ {1}}} \ Rightarrow AC = AC_ {2} \ Rightarrow \ triangle ABC = \ triangle ABC_ {2}}

( first sign )

\Rightarrow \angle B=\angle ABC_{2}\Rightarrow \triangle ABC\sim \triangle A_{1}B_{1}C_{1}

{\ displaystyle \ Rightarrow \ angle B = \ angle ABC_ {2} \ Rightarrow \ triangle ABC \ sim \ triangle A_ {1} B_ {1} C_ {1}}

( first sign ).

Third Character

If the three sides of one triangle are respectively proportional to the three sides of the other, then such triangles are similar.

Given : $\triangle ABC$ ${\ displaystyle \ triangle ABC}$ and $\triangle A_{1}B_{1}C_{1},{\frac {AB}{A_{1}B_{1}}}$ ${\ displaystyle \ triangle A_ {1} B_ {1} C_ {1}, {\ frac {AB} {A_ {1} B_ {1}}}}$ = ${\frac {AC}{A_{1}C_{1}}}$ ${\ displaystyle {\ frac {AC} {A_ {1} C_ {1}}}}$ = ${\frac {BC}{B_{1}C_{1}}}$ ${\ displaystyle {\ frac {BC} {B_ {1} C_ {1}}}}$ .

Prove : $\triangle ABC\sim \triangle A_{1}B_{1}C_{1}.$ ${\ displaystyle \ triangle ABC \ sim \ triangle A_ {1} B_ {1} C_ {1}.}$

Evidence

1) Consider $\triangle ABC_{2}$ ${\ displaystyle \ triangle ABC_ {2}}$ , wherein $\angle BAC_{2}=\angle A_{1}$ ${\ displaystyle \ angle BAC_ {2} = \ angle A_ {1}}$ and $\angle ABC_{2}=\angle B_{1}:$ ${\ displaystyle \ angle ABC_ {2} = \ angle B_ {1}:}$

\triangle ABC_{2}\sim \triangle A_{1}B_{1}C_{1}

{\ displaystyle \ triangle ABC_ {2} \ sim \ triangle A_ {1} B_ {1} C_ {1}}

( first sign )

\Rightarrow {\frac {AB}{A_{1}B_{1}}}={\frac {AC_{2}}{A_{1}C_{1}}}.

{\ displaystyle \ Rightarrow {\ frac {AB} {A_ {1} B_ {1}}} = {\ frac {AC_ {2}} {A_ {1} C_ {1}}}.}

2) By condition:

{\frac {AB}{A_{1}B_{1}}}

{\ displaystyle {\ frac {AB} {A_ {1} B_ {1}}}}

=

{\frac {AC}{A_{1}C_{1}}}

{\ displaystyle {\ frac {AC} {A_ {1} C_ {1}}}}

=

{\frac {BC}{B_{1}C_{1}}}\Rightarrow \

{\ displaystyle {\ frac {BC} {B_ {1} C_ {1}}} \ Rightarrow \}

AC = AC ₂ , BC = BC ₂ => ∆ABC = ∆ABC ₂ ( third symptom );
∆ABC ₂

\sim

{\ displaystyle \ sim}

∆A ₁ B ₁ C ₁ =>

\triangle ABC\sim \triangle A_{1}B_{1}C_{1}

{\ displaystyle \ triangle ABC \ sim \ triangle A_ {1} B_ {1} C_ {1}}

.

Signs of similarity of right triangles

On an acute angle - see the first sign ;
For two legs - see the second symptom ;
For leg and hypotenuse - see the third symptom .

Properties of similar triangles

The area ratio of such triangles is equal to the square of the similarity coefficient
The volume ratio of such stereometric figures is equal to the cube of the similarity coefficient
The ratio of perimeters and lengths of bisectors , medians , heights and mid-perpendiculars is equal to the similarity coefficient.

Examples of similar triangles

The following types of triangles are similar:

The complementary triangle and anti-complementary triangle are similar; their respective sides are parallel.
Triangle ABC is similar to its complementary triangle ; their respective sides are parallel and relate as 2: 1.
The triangle ABC is similar to its anti-complementary triangle ; their respective sides are parallel and relate as 1: 2.
Source triangle $\Delta ABC$ ${\ displaystyle \ Delta ABC}$ with respect to the orthogon triangle is a triangle of three external bisectors ^[1] .
The ortho-triangle and tangential triangle are similar (Zötel, Corollary 1, § 66, p. 81).
The ortho - triangle of the ortho- triangle and the original triangle are similar.
The triangle of the three outer bisectors of the triangle of the three outer bisectors and the original triangle are similar.
Suppose that the tangent points of the circle inscribed in the given triangle are connected by segments, then we get the Gergonn triangle , and the heights are drawn in the resulting triangle. In this case, the straight lines connecting the bases of these heights are parallel to the sides of the original triangle. Therefore, the ortho- triangle of the Gergonn triangle and the original triangle are similar.
The above mentioned similarity properties of related triangles are a consequence of the listed properties of the parallelism of the sides of related triangles .
Theorem : a circumference-chevian triangle is similar to a podder ^[2] . The definitions used here are:
- A triangle with vertices at the second points of intersection of lines drawn through the vertices and a given point, with a circumscribed circle, is called a circle-chevron triangle .
- A triangle with vertices in the projections of a given point on the sides is called a podnoy or pedal triangle of this point.

The properties of parallelism (antiparallelism) of the sides of related triangles

The corresponding sides of the additional triangle , anti-complementary triangle and the original triangle are pairwise parallel.
The sides of this acute-angled triangle are antiparallel to the corresponding sides of the orthogon triangle against which they lie.
The sides of the tangential triangle are antiparallel to the corresponding opposite sides of the given triangle (by the antiparallel property of tangents to the circle).
The sides of the tangential triangle are parallel to the corresponding sides of the ortho- triangle .
Suppose that the tangent points of the circle inscribed in the given triangle are connected by segments, then we get the Gergonn triangle , and the heights are drawn in the resulting triangle. In this case, the straight lines connecting the bases of these heights are parallel to the sides of the original triangle. Therefore, the ortho- triangle of the Gergonn triangle and the original triangle are similar.

Similarity in a Right Triangle

Triangles by which a height omitted from a right angle divides a right-angled triangle are similar to the entire triangle by the first sign , which means:

The height of the right-angled triangle, lowered to the hypotenuse, is equal to the geometric mean of the projections of the legs to the hypotenuse ,
The leg is equal to the geometric mean hypotenuse and the projection of this leg on the hypotenuse.

Related Definitions

The similarity coefficient is the number k equal to the ratio of the similar sides of similar triangles.
The similar sides of such triangles are the sides lying opposite equal angles.

Notes

↑ Starikov V. N. Geometry Research // Collection of publications of the Globus scientific journal based on the materials of the Vth International Scientific and Practical Conference “Achievements and Problems of Modern Science”, St. Petersburg: collection with articles (standard level, academic level). S-P .: Scientific journal Globus , 2016.S. 99-100
↑ The system of problems in geometry of R. K. Gordin. Task 6480

Literature

Geometry 7-9 / L. S. Atanasyan et al. - 12th ed. - M .: Education, 2002 .-- 384 p.:
Zetel S.I. New geometry of a triangle. A manual for teachers. 2nd edition. M.: Uchpedgiz, 1962.153 s.

Links

[1] Starikov V. N. Geometry Research // Collection of publications of the Globus scientific journal based on the materials of the Vth International Scientific and Practical Conference “Achievements and Problems of Modern Science”, St. Petersburg: collection with articles (standard level, academic level). S-P .: Scientific journal Globus , 2016.S. 99-100

[2] The system of problems in geometry of R. K. Gordin. Task 6480

Signs of similarity of triangles

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