Homotheties and Similarities A file of the Geometrikon gallery by Paris Pamfilos We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time. T.S. Eliot, Little Gidding Contents (Last update: 30‑05‑2021) 1 Homotheties 1 2 Homotheties and triangles 2 3 Homotheties with different centers 4 4 Homotheties and translations 5 5 Representation and group properties of homotheties 6 6 Similarities, general definitions 7 7 Similarities defined by two segments 9 8 Similarities and orientation 13 9 Similarities and triangles 14 10 Triangles varying by similarity 16 11 Relative position in similar figures 18 12 Polygons on the sides of a triangle 19 13 Representation and group properties of similarities 24 14 Logarithmic spiral and pursuit curves 24 1 Homotheties Homotheties of the plane and their generalization, the “similarities”, to be discussed be‑ low, are transformations which generalize those of “isometries” or “congruences” of the plane (see file Isometries). Given a number 휅 ≠ 0 and point 푂 of the plane, we call “homothety” of “center” 푂 and “ratio” 휅 the transformation which corresponds: a) to point 푂, itself, b) to every point 푋 ≠ 푂 the point 푋′ on the line 푂푋, such that the following signed ratio relation holds: 푂푋′ = 휅. 푂푋 A direct consequence of the definition is, that for every point 푂 the homothety of center 2 Homotheties and triangles 2 O X X' Figure 1: Homothety 푂 and ratio 휅 = 1 is the identity transformation. Often, when the ratio is 휅 < 0 we say that the transformation is an “antihomothety”. Its characteristic is that point 푂 is between 푋 and 푋′. Theorem 1. The composition of two homotheties with center 푂 and ratios 휅 and 휆 is a homothety of center 푂 and ratio 휅 ⋅ 휆. Proof. Obvious consequence of the definition. If 푓 and 푔 are the two homotheties with the same center 푂 and ratios respectively 휅 and 휆, then, for every point 푋, points 푌 = 푓 (푋), 푍 = 푔(푌) and 푂 will be four points on the same line and will satisfy, 푂푌 푂푍 푂푍 푂푍 푂푌 = 휅, = 휆 ⇒ = ⋅ = 휆 ⋅ 휅 . 푂푋 푂푌 푂푋 푂푌 푂푋 Corollary 1. The inverse transformation of a homothety 푓 , of center 푂 and ratio 휅, is the homothety 1 with the same center and ratio 휅 . Remark 1. The homothety is a special transformation closely connected with Thales’ the‑ orem and the “similarity of triangles”, i.e “triangles which have equal corresponding angles” 퐿푒푓 푡푟푖푔ℎ푡푎푟푟표푤 “triangles which have proportional corresponding sides”. Two triangles, and more general two shapes {Σ, Σ′} are called “homothetic” when there exists a homothety 푓 mapping one onto the other 푓 (Σ) = Σ′. Exercise 1. Find all homotheties that transform a given point 푋 to another point 푋′ ≠ 푋. Homothetic triangles are particular cases of “similar triangles” and are the key to in‑ vestigate properties of more general homothetic shapes. 2 Homotheties and triangles Theorem 2. A homothety 푓 with center 푂 maps a triangle 푂푋푌 to a similar triangle 푂푋′푌′ with {푋′ = 푓 (푋), 푌′ = 푓 (푌)}, the sides {푋푌, 푋′푌′} being parallel. Proof. A simple application of Thales’ theorem. Corollary 1. A homothety maps a line 휀 to a parallel line 휀′. Exercise 2. Find all homotheties that transform a given line 휀 to a given parallel to it 휀′. Distin‑ guish the cases {휀 ≠ 휀′ , 휀 = 휀′}. Theorem 3. A homothety maps a triangle 퐴퐵퐶 to a similar triangle 퐴′퐵′퐶′. Proof. Application of the previous corollary and Thales’ theorem. Corollary 2. A homothety 푓 preserves the angles and multiplies the distances between points with its ratio. In other words, for every pair of points 푋, 푌 and their images 푋′ = 푓 (푋), 푌′ = 푓 (푌) holds |푋′푌′| = 휅|푋푌| and for every three points the respective angles are preserved 푌′̂푋′푍′ = 푌푋푍̂ . 2 Homotheties and triangles 3 Exercise 3. Show that there is no genuine homothety with ratio 푘 ≠ 1 mapping a triangle to itself. Theorem 4. Two triangles 퐴퐵퐶 and 퐴′퐵′C′, which have their corresponding sides parallel are similar. C C 1 C' Α Β Β Β 2 1 Α' Β' C 2 Figure 2: Triangles with corresponding sides parallel Proof. Translate triangle 퐴′퐵′C′ and place it in such a way, that the vertices 퐴 and 퐴′ coincide and the lines of their sides 퐴퐵, 퐴C coincide respectively with 퐴′퐵′ and 퐴′C′ (See Figure 2). The translated triangle will take the position 퐴퐵1C1 or 퐴퐵2C2, with its third side parallel to 퐵C. Therefore, it will be similar to 퐴퐵퐶, while it is also congruent to the initial 퐴′퐵′C′. Theorem 5. For two triangles 퐴퐵퐶 and 퐴′퐵′C′, which have parallel corresponding sides, the lines 퐴퐴′, 퐵퐵′ and CC′, which join the vertices with the corresponding equal angles, either pass through a common point and the triangles are homothetic, or are parallel and the triangles are congruent. A' A A B O B' C' B' C B O C' C A' Figure 3: Homothetic triangles Proof. Let 푂 be the intersection point of 퐴퐴′ and 퐵퐵′. We will show that CC′ also passes |퐴퐵| |푂퐴| |푂퐵| through point 푂. According to Thales, we have equal ratios |퐴′퐵′| = |푂퐴′| = |푂퐵′| =휅. Con‑ C C″ |푂C| ′ ′C″ sider therefore on 푂 point with |푂C″| =휅. The created triangle 퐴 퐵 has sides pro‑ portional to those of 퐴퐵퐶, therefore it is similar to it and consequently has the same angles. It follows, that 퐴′퐵′C′ and 퐴′퐵′C″ have 퐴′퐵′ in common and same angles at 퐴′ and 퐵′, therefore they coincide and C′ = C″, in other words, 푂C passes through C′ too. This reasoning shows also that, if the two lines 퐴퐴′ and 퐵퐵′ do not intersect, that is if they are parallel, then the third line will also be necessarily parallel to them and 퐴퐵퐵′퐴′, 퐵CC′퐵′ and 퐴CC′퐴′ will be parallelograms, therefore the triangles will have correspond‑ ing sides equal. 3 Homotheties with different centers 4 3 Homotheties with different centers Theorem 6. The composition of two homotheties 푓 and 푔 with different centers 푂 and 푃 and ratios respectively 휅 and 휆, with 휅 ⋅ 휆 ≠ 1, is a homothety with center 푇 on the line 푂푃 and ratio equal to 휅 ⋅ 휆. Proof. The proof is an interesting application of Menelaus’ theorem (see file Menelaus’ theorem). Let 푋 be an arbitrary point and 푌 = 푓 (푋), 푍 = 푔(푌). This defines the triangle 푂푌푃 and the points 푋, 푍 are contained in its sides 푂푌 and 푌푃 respectively. Let 푇 be the intersection point of 푍푋 with 푂푃. Applying Menelaus’ theorem we have, Z Y X T P O Figure 4: Composition of homotheties with 휅휆 ≠ 1 푋푂 푍푌 푇푃 푇푃 푋푌 푍푃 ⋅ ⋅ = 1 ⇒ = ⋅ . 푋푌 푍푃 푇푂 푇푂 푋푂 푍푌 However, for the oriented line segments holds 푋푌 푋푂 + 푂푌 푂푌 푋푌 = 푋푂 + 푂푌 ⇒ = = 1 + = 1 − 휅, 푋푂 푋푂 푋푂 푍푌 푍푃 + 푃푌 푃푌 1 푍푌 = 푍푃 + 푃푌 ⇒ = = 1 + = 1 − ⇒ 푍푃 푍푃 푍푃 휆 푇푃 푋푌 푍푃 1 휆 ⋅ (1 − 휅) = ⋅ = (1 − 휅) ⋅ ⎜⎛ ⎟⎞ = . 푇푂 푋푂 푍푌 1 휆 − 1 ⎝1 − 휆 ⎠ The last formula shows, that the position of 푇 on the line 푂푃 is fixed and independent 푇푍 of 푋. In addition, the ratio 휇 = 푇푋 is calculated, by applying Menelaus’ theorem to the triangle 푂푋푇, this time with 푃푌 as secant: 푃푇 푍푋 푌푂 ⋅ ⋅ = 1 ⇒ 푃푂 푍푇 푌푋 푍푋 푌푋 푃푂 = ⋅ ⇔ 푍푇 푌푂 푃푇 푍푇 + 푇푋 푌푂 + 푂푋 푃푇 + 푇푂 = ⋅ ⇔ 푍푇 푌푂 푃푇 1 푂푋 푇푂 1 − = (1 + ) (1 + ) ⇔ 휇 푌푂 푃푇 1 1 휆 − 1 1 − = (1 − ) (1 − ) ⇔ 휇 휅 휆(1 − 휅) 휇 = 휅휆 . Theorem 7. The composition of two homotheties 푓 and 푔 with different centers 푂 and 푃 respec‑ tively and ratios 휅 and 휆 with 휅 ⋅ 휆 = 1 is a translation by interval parallel to 푂푃. 4 Homotheties and translations 5 Y Z X P O Figure 5: Composition of homotheties with 휅휆 = 1 Proof. Let 푋 be an arbitrary point and 푌 = 푓 (푋), 푍 = 푔(푌). This defines the triangle 푂푌푃 and the points 푋, 푍 are contained in its sides 푂푌 and 푌푃 respectively. According to the hypothesis 푌푋 푌푂 + 푂푋 1 푌푍 푌푃 + 푃푍 푃푍 1 = = 1 − , = = 1 − = 1 − 휆 = 1 − . 푌푂 푌푂 휅 푌푃 푌푃 푌푃 휅 The equality of the ratios shows, that the line segment 푋푍 is parallel to 푂푃. From the similarity of triangles 푌푂푃 and 푌푋푍, follows that 푋푍 = (1 − 휆)푂푃, therefore 푋푍 has fixed length and direction. 4 Homotheties and translations Theorem 8. The composition 푔 ∘ 푓 of a homothety and a translation 푔 is a homothety. Proof. Assume that the homothety has center 푂 and ratio 휅 and the translation is defined by the fixed, oriented line segment 퐴퐵. Let also 푋 ≠ 푂 be arbitrary and 푌 = 푓 (푋), 푍 = 푔(푌). Assume finally that 푃 is the intersection of the line 푋푍 and the line 휀 is the parallel Y A B Z X ε P O Figure 6: Compositions of homotheties and a translation to 퐴퐵 from 푂 (See Figure 6). From the similarity of the triangles 푋푌푍 and 푂푋푃 follows that 푂푃 푂푃 푂푋 푂푋 1 1 1 = = = = = ⇒ 푂푃 = 퐴퐵. 퐴퐵 푌푍 푌푋 푌푂 + 푂푋 푌푂+푂푋 1 − 휅 1 − 휅 푂푋 It follows that the position of 푃 on 휀 is fixed and independent of 푋. Also for the ratio, 푃푍 푂푌 = = 휅. 푃푋 푂푋 Therefore the composition 푔 ∘ 푓 is a homothety of center 푃 and ratio 휅.