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Cross Ratio A file of the Geometrikon gallery by Paris Pamfilos

. . . it is hard to communicate understanding because that is something you get by living with a problem for a long time. You study it, perhaps for years, you get the feel of it and it is in your bones. You can’t convey that to anybody else. Having studied the problem for five years you may be able to present it in such a way that it would take somebody else less time to get to that than it took you but if they haven’t struggled with the problem and seen all the pitfalls, then they haven’t really understood it.

Interview with M. Atiya. Math. Intelligencer 6(1984)

Contents (Last update: 09‑02‑2021)

1 Cross Ratio 2 2 Properties of cross‑ratio of four numbers 3

3 Cross ratio expressed through angles 4

4 Harmonic pencils of lines 6

5 Coincidences and cross ratios of pencils 8

6 Cross ratio on a circle 9 7 Cross Ratio on a conic 11 8 Cross Ratio Formularium 12 9 Complex Cross Ratio 12

10 Circle parametrization through the cross ratio 14

11 Further properties of the cross ratio 14 1 Cross Ratio 2

1 Cross Ratio

The cross ratio of four points {퐴, 퐵, 퐶, 퐷} of a 휀 of the “euclidean ” is defined by the quotient of two signed ratios of distances ∶ 퐶퐴 퐷퐴 (퐴퐵, 퐶퐷) = ∶ . (1) 퐶퐵 퐷퐵 As can be seen in the file , this definition is compatible with the “projective” one given there. Notice also that the whole number system and the cross ratio can be defined by simple geometric constructions based onthe “axioms of ” (see [VY10, I,p.141]). Analogously to this definition we can select another ordering of the letters and define different similar quotients of ratios. There result 4! = 24 permuted symbols, giving by 4 the same values as in the following table. (퐴퐵, 퐶퐷) = (퐵퐴, 퐷퐶) = (퐷퐶, 퐵퐴) = (퐶퐷, 퐴퐵) = 휆 (퐴퐵, 퐷퐶) = (퐵퐴, 퐶퐷) = (퐶퐷, 퐵퐴) = (퐷퐶, 퐴퐵) = 1/휆 (퐴퐶, 퐵퐷) = (퐶퐴, 퐷퐵) = (퐷퐵, 퐶퐴) = (퐵퐷, 퐴퐶) = 1 − 휆 (퐴퐶, 퐷퐵) = (퐶퐴, 퐵퐷) = (퐵퐷, 퐶퐴) = (퐷퐵, 퐴퐶) = 1/(1 − 휆) (퐴퐷, 퐵퐶) = (퐷퐴, 퐶퐵) = (퐶퐵, 퐷퐴) = (퐵퐶, 퐴퐷) = (1 − 휆)/휆 (퐴퐷, 퐶퐵) = (퐷퐴, 퐵퐶) = (퐵퐶, 퐷퐴) = (퐶퐵, 퐴퐷) = 휆/(1 − 휆) Identifying the point with its line coordinate relative to an arbitrary coordinate system of

O E A B C D

Figure 1: Four points on a line the line, i.e. fixing two points {푂, 퐸} on the line and measuring the location of an arbitrary point 푋 by the signed ratio 푥 = 푂푋/푂퐸 (may be negative), the points {퐴, 퐵, 퐶, 퐷} define the respective coordinates {푎, 푏, 푐, 푑} and the cross ratio can be expressed through the “cross ratio of four numbers” 푎 − 푐 푎 − 푑 (퐴퐵, 퐶퐷) = ∶ . (2) 푏 − 푐 푏 − 푑 The expression on the right is independend of the location of points {푂, 퐸} i.e. indepen‑ dent of the special line coordinate system. In fact, changing the coordinate system to some other (defined by two other {푂′, 퐸′}), the new coordinate 푥′ is connected to the old 푥 by a relation of the form 푥 = 푚 ⋅ 푥′ + 푛 , for appropriate constants {푚, 푛}. The fact is that by substituting this expression of 푥 into the formula we get the same number expressed in the other coordinates i.e. 푎′ − 푐′ 푎′ − 푑′ 푎 − 푐 푎 − 푑 ∶ = ∶ . (3) 푏′ − 푐′ 푏′ − 푑′ 푏 − 푐 푏 − 푑 More generally the cross ratio of four numbers, defined by equation 3 is under the “broken linear transformations” or “Moebius transformations” (see file Projective line)∶ 푥 = 푓 (푥) = (푚 ⋅ 푥′ + 푛)/(푝 ⋅ 푥′ + 푞) with 푚푞 − 푛푝 ≠ 0. (4) This makes it possible to generalize the cross ratio for four points {퐴, 퐵, 퐶, 퐷} lying on a conic. Relations between two variables expressed through equation 4 are called “homo‑ graphic relations” and the functions 푓 are called “”. Latter are subject ofa further study in the files Projective line and Homographic Relation. 2 Properties of cross‑ratio of four numbers 3

The definition allows for a number to be taken at infinity. For example, taking 퐷(푑) at infinity (푑 = ∞), the cross ratio reduces to the signed ratio

(퐴퐵, 퐶∞) = (푎푏, 푐∞) = (푎 − 푐)/(푏 − 푐) = 퐶퐴/퐶퐵. (5)

Taking further 푐 = 0, this reduces to 푎/푏. Thus, the ratio of two numbers is the cross ratio (푎푏, 0∞). Using a we can transform a cross ratio (푎푏, 푐푑) to an equal one of the form (푎′푏′, 푐′∞) . This often simplifies proofs involving cross ratios. An example is the following [Ber87, I,p.138]. Exercise 1. The following equation is valid for five arbitrary points {퐴, 퐵, 퐶, 푈, 푉} on a line∶

(퐴퐵, 푈푉)(퐵퐶, 푈푉)(퐶퐴, 푈푉) = 1. (6)

The property becomes trivial by setting 푉 = ∞ ∶ 푈퐴 푈퐵 푈퐶 (퐴퐵, 푈∞)(퐵퐶, 푈∞)(퐶퐴, 푈∞) = ⋅ ⋅ = 1. (7) 푈퐵 푈퐶 푈퐴 It is astonishing how many geometric properties depend on the cross‑ratio. In fact, it can be proved that the cross ratio, considered as a of four collinear points (variables), is the unique invariant of the projective geometry. Hence one can expect to find it behind every incidence relation between points and lines. Exercise 2. Show that, given three different points {퐴, 퐵, 퐶} on the line 휀, the position of a fourth point 푋 on 휀 is completely determined through the cross ratio 푘 = (퐴퐵, 퐶푋).

2 Properties of cross‑ratio of four numbers

Next list of properties of the cross ratio gives, among other things, the reason for the validity of the relations between the 24 possible orderings of the table of the preceding section. The first two properties are discussed in the file Projective line. The rest is proved by easy calculations.

1. Given three pairwise different real numbers {푥1, 푥2, 푥3} the “cross ratio function” 푦 = 푓 (푥) = (푥1푥2, 푥3푥) defines an invertible “homographic relation”, whose graph is a rectangular hyperbola.

2. Given three pairs of real numbers {(푥1, 푦1), (푥2, 푦2), (푥3, 푦3)} in general position, the equation (푦1푦2, 푦3푦) = (푥1푥2, 푥3푥) solved for 푦 defines uniquely a homographic re‑ lation 푦 = 푓 (푥), such that {푓 (푥1) = 푦1, 푓 (푥2) = 푦2, 푓 (푥3) = 푦3.} 3. Every permutation, which is product of two “transpositions” of the letters {푎, 푏, 푐, 푑} leaves the cross ratio invariant i.e.

(푎푏, 푐푑) = (푏푎, 푑푐) = (푐푑, 푎푏) = (푑푐, 푏푎).

Hence from the 4! in total permutations of the 4 letters, the various resulting cross ratios obtain only 6 different values (seen in the table of section 1).

4. (푎푏, 푑푐) = (푎푏, 푐푑)−1 and (푎푐, 푏푑) + (푎푏, 푐푑) = 1. From these follow the last equalities in the rows of the table of section 1.

5. The four numbers 푎, 푏, 푐, 푑 are pairwise different if and only if the cross ratio (푎푏, 푐푑) has a value different from 1, 0, ∞. 3 Cross ratio expressed through angles 4

A special, but important case of cross ratio is that, which results from two points (퐴, 퐵) and two others (퐶, 퐷), which are harmonic conjugate to the first and for which we say that they are “harmonic pairs” and denoted by (퐴, 퐵) ∼ (퐶, 퐷). This, by definition means, that the signed ratios are the same, except for the sign, 퐶퐴 퐷퐴 퐶퐴 퐷퐴 = 휆, = −휆 ⇒ ∶ = −1. 퐶퐵 퐷퐵 퐶퐵 퐷퐵 In this case the 6 values of the various cross ratios reduce to three ∶ {−1, 1/2, 2} and we say also that {퐶, 퐷} are “harmonic conjugate to {퐴, 퐵}”. We say also that 퐷 is “harmonic conjugate to 퐶 ” w.r. to {퐴, 퐵} and denote this by the symbol 퐷 = 퐶(퐴퐵). Obviously for fixed {퐴, 퐵} this relation between {퐶, 퐷} is symmetric 퐷 = 퐶(퐴퐵) ⇔ 퐶 = 퐷(퐴퐵).

3 Cross ratio expressed through angles

Take four points {퐴, 퐵, 퐶, 퐷} on a line 휀 and a fifth point 퐸 outside 휀. Then the cross ratio has an interpretation in terms of the angles at 퐸 of lines {퐸퐴, 퐸퐵, 퐸퐶, 퐸퐷} (See Figure 2).

DA

E β* α β α* ε C A C B D B C A DB

Figure 2: Cross ratio (퐴퐵, 퐶퐷) in terms of angles

We speak about the “ of lines” at 퐸 and denote it with 퐸(퐴퐵퐶퐷) and also denote the corresponding cross ratio with 퐸(퐴퐵, 퐶퐷) . Projecting the points {퐶, 퐷} on the lines {퐸퐴, 퐸퐵} and doing some simple calculations suggested by figure 2, we find the relations ∗ 퐴퐶 퐴퐷 퐴퐶 퐵퐷 퐶퐴퐶 퐷퐵퐷 퐸퐶 sin(훼) 퐸퐷 sin(훼 ) ∶ = ⋅ = ⋅ = ∗ ⋅ 퐵퐶 퐵퐷 퐴퐷 퐵퐶 퐷퐴퐷 퐶퐵퐶 퐸퐷 sin(훽 ) 퐸퐶 sin(훽) Thus, finally sin(훼) sin(훽) (퐴퐵, 퐶퐷) = ∶ . (8) sin(훼∗) sin(훽∗) The dependence of 퐸(퐴퐵, 퐶퐷) only from the angles, shows that keeping the four lines {퐸퐴, 퐸퐵, 퐸퐶, 퐸퐷} fixed and varying the line 휀 , the cross ratio defined by the four intersec‑ tions on 휀 does not change its value. Also if we vary the position of 퐸 so that the angles between the four lines remain fixed, then the cross ratio on any line 휀 defined by the four intersections has a constant value. These simple observations lead to the following theorems. Theorem 1. A pencil of four lines 퐸(퐴퐵퐶퐷) intersects on any line 휀 four points whose cross ratio 퐸(퐴퐵, 퐶퐷) is constant and independent of the particular position of the intersecting line 휀 (See Figure 3). 3 Cross ratio expressed through angles 5

E ε'

C' D' A' B'

ε ABCD

Figure 3: Cross ratio of a pencil of four lines

Theorem 2. Fixing four points {퐴, 퐵, 퐶, 퐷} on a circle 휅 and varying a fifth point 퐸 ∈ 휅 the pencil 퐸(퐴퐵퐶퐷) defines on any line 휀 intersecting the pencil a cross ratio 퐸(퐴퐵, 퐶퐷) indepen‑ dent (i) from the particular position of 퐸 on the circle and (ii) independent also from the particular line intersecting the pencil (see figure 4).

The first theorem allows us to associate a number to each quadruple of linespassing through a common point denoted by 퐸(퐴퐵, 퐶퐷). This is the cross ratio, which the four lines define on any intersecting them line. According to the theorem, this cross ratiois independent of the special intersecting line and, consequently, defines some characteris‑ tic of the four lines. We call this number “Cross ratio of four lines” which pass through the

E

κ

D A

B C ε A' B' C' D'

Figure 4: Cross ratio of four points on a circle same point, or cross ratio of the pencil of four lines. A pencil consisting of four lines whose cross ratio equals 퐸(퐴퐵, 퐶퐷) = −1 is called a “harmonic pencil ”. Analogously, last theorem is important in that it allows to speak about the “cross ra‑ tio (퐴퐵, 퐶퐷) of four points on a circle”. One has only to take an arbitrary fifth point 퐸 on the circle and an arbitrary line 휀 and apply the last theorem. Then define (퐴퐵, 퐶퐷) through (퐴′퐵′, 퐶′퐷′) , where {퐴′, 퐵′, 퐶′, 퐷′} are the intersections of the line 휀 with the lines {퐸퐴, 퐸퐵, 퐸퐶, 퐸퐷} (See Figure 4). Obviously the relation of “harmonic pairs” and points {퐶, 퐷} “harmonic conjugate” to {퐴, 퐵} extends to pairs of points on the circle.

Theorem 3. Consider four given lines {휂1, 휂2, 휂3, 휂4} passing through a fixed point 퐸 and ex‑ pressed as a linear combination of two other fixed lines {휇, 휈 ∶ 휂푖 = 휇 + 휆푖휈 , 푖 = 1, 2, 3, 4} through 퐸. Then, the cross ratio of the four lines is given by

휆1 − 휆3 휆1 − 휆4 (휂1휂2, 휂3휂4) = ∶ . 휆2 − 휆3 휆2 − 휆4 4 Harmonic pencils of lines 6

Proof. If the lines are {휇 ∶ 푎푥 + 푏푦 + 푐 = 0 , 휈 ∶ 푎′푥 + 푏′푦 + 푐′ = 0}, then

′ ′ ′ 휂푖 = 휇 + 휆푖휈 = (푎 + 휆푖푎 )푥 + (푏 + 휆푖푏 )푦 + (푐 + 휆푖푐 ) = 0.

Intersecting these lines with the x‑axis 휀 ∶ 푦 = 0 leads to the equations for the correspondig x: ′ ′ ′ 푐 + 휆푖푐 (푎 + 휆푖푎 )푥 + (푐 + 휆푖푐 ) = 0 ⇒ 푥푖 = − ′ . 푎 + 휆푖푎 푥 −푥 The cross ratio of the four lines is then 1 3 ∶ 푥1−푥4 , which by substitution of the {푥 } 푥2−푥3 푥2−푥4 푖 and simplification leads to the stated formula.

4 Harmonic pencils of lines

퐶퐴 Corollary 1. The cross ratio of four lines {퐸퐴, 퐸퐵, 퐸퐶, 퐸퐷} is equal to the signed ratio 퐶퐵 of the three points {퐴퐵퐶}, which the three first lines excise on a line 휀, which is parallel to the fourth line 휀′ = 퐸퐷.

E ε' D

ε A C B

Figure 5: Cross ratio of four lines on a parallel of 휀′

Proof. The proof follows from the comment made above, according to which, the cross 퐶퐴 ratio (퐴퐵, 퐶퐷) is reduced to the simple signed ratio 퐶퐵 , in the case where the fourth point 퐷 tends to infinity (See Figure 5).

Corollary 2. A pencil of four lines 퐸(퐴퐵퐶퐷) is harmonic, if and only if a parallel 휀 to the fourth line 퐸퐷 intersects the others at three points {퐴, 퐵, 퐶} of which one (퐶) is the middle of the line segment (퐴퐵) which is defined by the other two.

Next proposition and the one after that examine the two main examples of harmonic pencils.

O

D A C B ΧΥ

Figure 6: Harmonic pencil of an angle and its bisectors

Theorem 4. In every triangle 푂퐴퐵 the two sides 푂퐴, 푂퐵 and the bisectors 푂퐶, 푂퐶 of the angle at 푂 make a harmonic pencil. 4 Harmonic pencils of lines 7

Proof. It is well known that in the triangle 푂퐴퐵 with bisectors 푂퐶 and 푂퐷, holds

|퐶퐴| |퐷퐴| |푂퐴| 퐶퐴 퐷퐴 = = ⇒ ∶ = −1. |퐶퐵| |퐷퐵| |푂퐵| 퐶퐵 퐷퐵

Theorem 5. Show that if in the harmonic pencil 푂(푋푌푍푊) the two lines {푂푋, 푂푌} are orthog‑ onal, then these coincide with the bisectors of the angle 푍푂푊̂ .

Proof. Assume that the intersection points of the pencil with a line 휀 satisfy (푋푌, 푍푊) = −1. Consider the harmonic conjugate 푂푊′ of 푂푍 relative to 푂푋 (See Figure 7). Then

O

ε W' Υ Z Χ W

Figure 7: Harmonic pencil with bisectors the corresponding point 푊′ on 휀 will satisfy also (푋푌, 푍푊′) = −1, therefore it will be coincident with point 푊.

Theorem 6. In every triangle 푂퐸퐵 the two sides {푂퐸, 푂퐵}, the median 푂푀 and the parallel to the base 퐸퐵 from 푂 make a harmonic pencil.

O D A

C EB M

Figure 8: Harmonic pencil of median and opposite side

Proof. If we draw in triangle 푂퐸퐵 the median 푂푀 and we extend the other median 퐵퐴 until it intersects the parallel to the base at 퐷 (see figure 8), this forms a pencil of lines 푂(퐴퐵퐶퐷) and holds 퐶퐴 퐷퐴 1 − = = , 퐶퐵 퐷퐵 2 from which follows that the pencil is harmonic (alternatively apply corrolary 2).

Next is a key figure containing several harmonic pairs and being thus useful inmany applications in euclidean, affine and projective geometry.

Theorem 7. In the complete quadrilateral of figure 9 the pairs (퐴, 퐵) ∼ (퐺, 퐻) are harmonic. 5 Coincidences and cross ratios of pencils 8

J C F Ι E Κ D H ABG

Figure 9: Harmonic quadruples and pencils in a complete quadrilateral

Proof. Apply Menelaus’ theorem to triangle 퐴퐵퐶 with secant 퐻퐾 and Ceva’s theorem to the same triangle relative to the point 퐷 ∶ 퐻퐴 퐹퐵 퐸퐶 퐸퐶 퐺퐴 퐹퐵 Menelaus: ⋅ ⋅ = 1, Ceva: ⋅ ⋅ = −1. 퐻퐵 퐹퐶 퐸퐴 퐸퐴 퐺퐵 퐹퐶 The result follows by dividing these relations side by side.

Corollary 3. All the quadruples of points lying on each line of figure 9 define harmonic pairs. Proof. From the pencil 퐶(퐴퐵퐺퐻) we find the relations (퐸, 퐹) ∼ (퐾퐻), (퐸, 퐵) ∼ (퐷, 퐼) and (퐴, 퐹) ∼ (퐷, 퐽). From the pencil 퐴(퐸퐵퐷퐼) follows analogously (퐾, 퐺) ∼ (퐷, 퐶).

5 Coincidences and cross ratios of pencils

Here we examine two coincidence relations, of three points on a line and three lines on a point, connected with the cross ratio of pencils. Theorem 8. Assume that the two line pencils 푂(훼훽훾훿), 푃(훼훽′훾′훿′) have the same cross ratio and common the line 훼, then the other corresponding lines intersect at three points 퐵 = 훽 ∩ 훽′, 퐶 = 훾 ∩ 훾′, 퐷 = 훿 ∩ 훿′ which are collinear.

Ο

β γ δ ε Α Β C D

α β' γ' δ'

Ρ

Figure 10: Pencils with common line and the same cross ratio

Proof. Consider the line 휀 = 퐵퐶 and the intersection point of 퐴 with 훼 (See Figure 10). On 휀 the two pencils will define the same cross ratio. Then if 퐷′ = 휀 ∩ 훿, 퐷″ = 휀 ∩ 훿′, then from the equality of the cross ratios (퐴퐵, 퐶퐷′) = (퐴퐵, 퐶퐷″), which the two pencils on 휀 define, follows that 퐷′ = 퐷″, therefore also the third intersection point 퐷′ = 퐷″ = 퐷 of 훿, 훿′ will be on 휀.

Theorem 9. Assume that on two lines {훼, 훽} intersecting at point 푂, are defined respectively points {퐴, 퐵, 퐶} and {퐴′, 퐵′, 퐶′}, such that the cross ratios (푂퐴, 퐵퐶), (푂퐴′, 퐵′퐶′) are equal. Then the lines {퐴퐴′, 퐵퐵′, 퐶퐶′} pass through a common point or are parallel (See Figure 10). 6 Cross ratio on a circle 9

Ο' β Β' C' Α' α Ο ΑΒC

Figure 11: Common cross ratio (푂퐴, 퐵퐶) = (푂퐴′, 퐵′퐶′)

Proof. Consider the case in which 푂′ = 퐴퐴′ ∩ 퐵퐵′ is a real point of the plane and draw the line 푂′퐶 intersecting the line 훽 at 퐶″ for which (푂퐴, 퐵퐶) = (푂퐴′, 퐵′퐶″). From the assumed (푂퐴, 퐵퐶) = (푂퐴′, 퐵′퐶′) we conclude 퐶′ = 퐶″. Analogous is the proof with par‑ allel lines, 푂′ is at infinity.

Exercise 3. Given is a triangle 푂퐵퐶 and three points {푋, 푌, 퐸} on its base 퐵퐶. From the point 퐸 are drawn lines intersecting the sides {푂퐶, 푂퐵} respectively at points {푋′, 푌′}. Show that the lines {푋푋′, 푌푌′} intersect at a point 푍 contained in a fixed line, which passes through 푂.

Χ' ζ

Ζ' Υ' Ο ε' Ζ ε ΒCYXED

Figure 12: Position determination from cross ratio

Hint: The line pencils 푂(퐵퐶퐷퐸) and 푍(푌푋퐷퐸) define the same cross ratio (푌′푋′, 푍′퐸) on line 푋′푌′, hence also on line 퐵퐶 (See Figure 12). Using a system of coordinates on the line 퐵퐶 the equation (퐵퐶, 퐷퐸) = (푋푌, 퐷퐸) contains only one unknown, the coordinate 푥 of 퐷. Another aspect of the last figure offers its connection with the following problem.

Problem 1. Given the four points {퐵, 퐶, 퐷, 퐸} on line 휀 and the variable point 푋 ∈ 휀 find the point 푌, such that the cross ratios are equal:

(푌푋, 퐷퐸) = (퐵퐶, 퐷퐸).

Hint: A geometric solution of the problem can be given by taking an arbitrary point 푂 and considering the pencil {푂(퐵퐶퐷퐸)} and an arbitrary line 휀′ through 퐸 (See Figure 12). The pencil defines on 휀′ the same cross ratio: 푂(퐵퐶, 퐷퐸) = (푌′푋′, 푍′퐸). The required point 푌 ∶ (푌푋, 퐷퐸) = (퐵퐶, 퐷퐸) defining the same cross ratio will have corresponding lines {푋푋′, 푌푌′, 퐷푍′} passing through the same point 푍 (theorem 9). From the given data we determine 푍 = 푋푋′ ∩ 퐷푍′ and drawing line 푌′푍 we find 푌 = 푌′푍 ∩ 휀.

6 Cross ratio on a circle

By a theorem discussed in the file Projective line, a variable tangent 휀 of a circle 휅 inter‑ sects on four other fixed tangents at the points {퐴, 퐵, 퐶, 퐷} of the circle, points {퐴′, 퐵′, 퐶′, 퐷′} 6 Cross ratio on a circle 10 in 휀 , whose cross ratio (퐴′퐵′, 퐶′퐷′) on 휀 is constant and independent of the particular position of 휀. It is interesting to note a consequence of this theorem, when the variable tangent 휀 takes the position of the tangent at one of the four points {퐴, 퐵, 퐶, 퐷}, like point 퐴

D X C ψD

ψC D' φ O D ε B φC C'

B' C Y 1 A'=A

D1

Figure 13: Cross ratio (퐴퐵, 퐶퐷) measured on the tangent at 퐴

′ ′ ′ say (See Figure 13). Then, projecting points {퐶 , 퐷 } to {퐶1, 퐷1} onto line 푂퐵 , the ratio (퐴′퐵′, 퐶′퐷′) becomes equal to

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ 퐶 퐴 퐷 퐴 퐶 퐴 퐷 퐵 푂퐶 sin(휙퐶) 푂퐷 sin(휓퐷) sin(휙퐶) sin(휓퐷) ′ ′ ∶ ′ ′ = ′ ′ ⋅ ′ ′ = ′ ⋅ ′ = ⋅ . 퐶 퐵 퐷 퐵 퐷 퐴 퐶 퐵 푂퐷 sin(휙퐷) 푂퐶 sin(휓퐶) sin(휙퐷) sin(휓퐶) Taking the points 푋 ∈ 퐴퐷′ and 푌 on the circle and comparing angles, we find that the last product of ratios is equal to

sin(푋퐴퐶)̂ sin(퐵퐴퐷)̂ sin(퐴푌퐶)̂ sin(퐵푌퐷)̂ ⋅ = ⋅ . sin(푋퐴퐷)̂ sin(퐵퐴퐶)̂ sin(퐴푌퐷)̂ sin(퐵푌퐶)̂ This shows that the cross ratio defined by the four tangents on a fifth tangent 휀 is equal to this expression, which is independent of the position of 푌 on the circle. The result is that the cross ratio (퐴′퐵′, 퐶′퐷′) is equal to the cross ratio of the four lines {푌퐴, 푌퐵, 푌퐶, 푌퐷}. By theorem 2 this is also the cross ratio (퐴퐵, 퐶퐷) of the four points on the circle. We obtain thus the following theorem.

Theorem 10. The cross ratio (퐴′퐵′, 퐶′퐷′) defined by the intersections with a tangent 휀 of four tangents of the circle at the points {퐴, 퐵, 퐶, 퐷} is equal to the cross ratio (퐴퐵, 퐶퐷) on the circle.

D'' ε CA B D C'' A'' B''

D' A' B' E κ C' O

Figure 14: Cross ratio of polars

Theorem 11. Four points on a line 휀 define the same cross ratio (퐴퐵, 퐶퐷) as their polars 퐸(퐴′퐵′, 퐶′퐷′) relative to the circle 휅. 7 Cross Ratio on a conic 11

Proof. The polars pass through the “pole” 퐸 of 휀 relative to 휅 (See Figure 14). The proof results by the equality of angles 퐴푂퐵̂ = 퐴″̂푂퐵″,… and the discussion in 3. Notice the equalities of the cross ratios defined by the polars on 휀.

(퐴퐵, 퐶퐷) = 퐸(퐴′퐵′, 퐶′퐷′) = (퐴″퐵″, 퐶″퐷″).

7 Cross Ratio on a conic

The cross ratio for points {푈, 푉, 푋, 푌} on a conic 휅 is defined by the same recipe used for circles. For this take an arbitrary line 휀 and an additional point 퐶 on the conic and define the “cross ratio of four points on the conic (푈푉, 푋푌) ” to be equal to the cross ratio (푈′푉′, 푋′푌′) (See Figure 15). This is independent of the position of 퐶 ∈ 휅 and also in‑

C U Y V X ε U' V' X' Y'

Figure 15: Cross of four points on a conic dependent from the particular line 휀 used in the definition. To see this one can usea “projectivity” mapping the conic onto the circle, which reduces the general case to that of the circle. By the same method of “” of a circle on an arbitrary conic is proved the anal‑ ogous to theorem 10 for conics:

ε V' U' X' Y'

U X

V Y

Figure 16: Cross of four points on a conic and tangents at these points

Theorem 12. The cross ratio (푈′푉′, 푋′푌′) defined by the intersections with a tangent 휀 of four tangents of the conic at the points {푈, 푉, 푋, 푌} is equal to the cross ratio (푈푉, 푋푌) on the conic (See Figure 16).

G κ A F

E B D ε C

Figure 17: Ratio (퐷퐸 ⋅ 퐹퐺)/퐸퐹 independent of position of 퐴 ∈ 휅 8 Cross Ratio Formularium 12

Next property, known as “Haruki’s lemma” is a direct implication of the definition of the cross ratio for points on a conic (See Figure 17).

Theorem 13. Given the fixed points {퐵, 퐶} on the conic 휅, the moving point 퐴 ∈ 휅 and the chord 퐷퐺 of 휅, the ratio (퐸퐷 ⋅ 퐺퐹)/퐸퐹 defined by the lines {퐴퐵, 퐴퐶} on the chord, is constant and independent of the position of 퐴 ∈ 휅.

Proof. By the definition of the cross ratio, (퐶퐺, 퐷퐵) = (퐸퐺, 퐷퐹) = (퐸퐷 ⋅ 퐺퐹)/(퐺퐷 ⋅ 퐸퐹) is constant and independent of the location of 퐴 ∈ 휅. But 퐺퐷 is constant.

8 Cross Ratio Formularium

In the following table we list the expressions of the cross ratio, relative to various coordi‑ nate systems, “projective” and “euclidean”. For the projective case and the appropriate for it “” we refer to the file Projective line.

1. For projective coordinates w.r. to the base {퐴, 퐵, 퐶} and 푃 = 푝1퐴 + 푝2퐵, … , 푝−푢 푝−푣 푝 = 푝1/푝2, 푞 = 푞1/푞2 … (푃푄, 푈푉) = 푞−푢 ∶ 푞−푣 .

2. Since {푎 = ∞, 푏 = 0, 푐 = 1 ∶} (퐴퐵, 푈푉) = 푣1 ∶ 푢1 (퐴퐵, 퐶푉) = 푣 = 푣1 . 푣2 푢2 푣2

푋퐴 푌퐴 3. For euclidean signed distance ratios: (퐴퐵, 푋푌) = 푋퐵 ∶ 푌퐵 . 푠 푡 4. For {푥 = (1 − 푠)푎 + 푠푏, 푦 = (1 − 푡)푎 + 푡푏}, (퐴퐵, 푋푌) = 푠−1 ∶ 푡−1 . 5. Measuring signed distances from the point 푂 of the line, the point 푋(푥) satisfying 푎푏(1−푘) (퐴퐵, 푂푋) = 푘 has 푥 = 푎−푘푏 . In particular, for the harmonic conjugate 푋0 of 푂 ∶ 2푎푏 푥0 = 푎+푏 is the “harmonic mean” of {퐴(푎), 퐵(푏)}.

sin([푐][푎]) sin([푑][푎]) 6. For oriented angles between lines {[푎], [푏], …} (퐴퐵, 퐶퐷) = sin([푐][푏] ∶ sin([푑][푏]) .

푓 (퐶) 푔(퐶) 7. For {푓 (푋) = 0, 푔(푋) = 0} representing lines {푂퐴, 푂퐵} (퐴퐵, 퐶푋) = 푓 (푋) ∶ 푔(푋) . In the last case we assume that lines {[푎] = 푂퐴, [푏] = 푂퐵} are given by equations corre‑ spondingly {푓 (푥, 푦) = 푝푥 + 푞푦 + 푟 = 0, 푔(푥, 푦) = 푝′푥 + 푞′푦 + 푟′ = 0.} For any point of the plane 푃 = (푝1, 푝2) denote by 푓 (푃) the number 푓 (푝1, 푝2), which is a multiple of the dis‑ tance of 푃 from the line [푓 ] ∶ 푓 (푥, 푦) = 0, i.e. 푓 (푃) = 푘 ⋅ 푑(푃, [푓 ]), with 푘 independent of the particular 푃. Thus, we have the following expression and its consequence:

푓 (퐶) 푑(퐶, [푓 ]) 푑(퐶, [푎]) 푑(퐶, [푏]) = ⇒ (퐴퐵, 퐶푋) = ∶ . 푓 (푋) 푑(푋, [푓 ]) 푑(푋, [푎]) 푑(푋, [푏])

9 Complex Cross Ratio

The “complex cross ratio” of four points in the is defined to be:

퐴 − 퐶 퐴 − 퐷 (퐴퐵, 퐶퐷) = ∶ , (9) 퐵 − 퐶 퐵 − 퐷 where points are identified with complex numbers ([Sch79, p.35]).

Theorem 14. The main properties of the complex cross ratio are the following: 9 Complex Cross Ratio 13

X

D

ε

A D'

Β C C' B' A'

Figure 18: Cross ratio of four complex numbers on a circle

1. (퐴퐵, 퐶퐷) is real if and only if the points are all four on a circle or on a line. 2. Assuming the points are on a circle, project them on a line 휀 from a point 푋 of the circle. Let {퐴′, 퐵′, 퐶′, 퐷′} be the corresponding projections. Then (퐴퐵, 퐶퐷) = (퐴′퐵′, 퐶′퐷′).

3. A “Moebius transformation” 푎푥 + 푏 푥′ = 푓 (푥) = with 푎, 푏, 푐, 푑 ∈ ℂ and 푎푑 − 푏푐 ≠ 0, (10) 푐푥 + 푑 preserves the complex cross ratio, i.e. if {푝′ = 푓 (푝), 푞′ = 푓 (푞), …}, then (푝′푞′, 푢′푣′) = (푝푞, 푢푣). (11)

4. Given two triples of complex numbers {(푝, 푞, 푟), (푝′, 푞′, 푟′)}, the equation (푝′푞′, 푟′푥′) = (푝푞, 푟푥) (12) defines the uniqe complex Moebius transformation 푥′ = 푓 (푥) mapping {푝, 푞, 푟} correspond‑ ingly to {푝′, 푞′, 푟′}. Nr‑1. For the proof write the complex numbers {퐴 − 퐶, 퐵 − 퐶, …} in polar form:

푖휙퐶퐴 푖휙퐶퐵 푖휙퐷퐴 푖휙퐷퐵 퐴 − 퐶 = 푟퐶퐴 ⋅ 푒 , 퐵 − 퐶 = 푟퐶퐵 ⋅ 푒 , 퐴 − 퐷 = 푟퐷퐴 ⋅ 푒 , 퐵 − 퐷 = 푟퐷퐵 ⋅ 푒 , implying

푟 ⋅ 푒푖휙퐶퐴 푟 ⋅ 푒푖휙퐷퐴 푟 푟 푒푖(휙퐶퐴−휙퐶퐵) 푟 푟 푒푖(퐴퐶퐵) (퐴퐵, 퐶퐷) = 퐶퐴 ∶ 퐷퐴 = 퐶퐴 ∶ 퐷퐴 = 퐶퐴 ∶ 퐷퐴 ⋅ , 푖휙퐶퐵 푖휙퐷퐵 푖(휙퐷퐴−휙퐷퐵) 푖(퐴퐷퐵) 푟퐶퐵 ⋅ 푒 푟퐷퐵 ⋅ 푒 푟퐶퐵 푟퐷퐵 푒 푟퐶퐵 푟퐷퐵 푒 where {(퐴퐶퐵), (퐴퐷퐵)} denotes the oriented angle‑measures. This quotient is real if and only if these angles are equal or complementary and inverse oriented, which shows the first claim. Nr‑2. The second claim follows from the previous one, since 푟 sin(퐴푋퐶)̂ 푟 sin(퐴푋퐷)̂ sin(퐴푋퐶)̂ sin(퐴푋퐷)̂ 퐶퐴 = , 퐷퐴 = ⇒ (퐴퐵, 퐶퐷) = ∶ , 푟퐶퐵 sin(퐵푋퐶)̂ 푟퐷퐵 sin(퐵푋퐷)̂ sin(퐵푋퐶)̂ sin(퐵푋퐷)̂ which according to theorem 1 is the expression of the cross ratio (퐴′퐵′, 퐶′퐷′). Nr‑3+nr‑4 are proved as suggested for the real case in the file Projective line. 10 Circle parametrization through the cross ratio 14

10 Circle parametrization through the cross ratio

Consider three fixed points {퐴, 퐵, 퐶} of the plane, identified with complex numbers. Then the equation (퐴퐵, 퐶푋) = 푡 with variable 푡 ∈ ℝ, (13) solved for 푋, defines a parametrization of the circumcircle of the triangle 퐴퐵퐶 by the real parameter 푡 ([Sch79, p.37]). Figure 19 illustrates this parametrization for a circle with

A

C

t B X

Figure 19: Parametrization of the circle through the cross ratio center at the origin. Points {퐴, 퐵, 퐶} map respectively to {∞, 0, 1} of the complex plane.

11 Further properties of the cross ratio

In the following list all points appearing in the formulas are on the same line 휀 and sym‑ bols such as {퐴퐵, 퐶퐷, …} represent signed distances of the points, except in the case of the symbol (퐴퐵, 퐶퐷) representing the cross ratio of the four collinear points. The formulas are to be found in [Cha52, ch.II] and [Pap96, ch.V].

1. If (퐴퐵, 퐶퐷) = (퐴퐵, 퐶′퐷′) then also (퐴퐵, 퐶퐶′) = (퐴퐵, 퐷퐷′). 1/퐴퐵 − 1/퐴퐷 2. One can write (퐴퐵, 퐶퐷) = . 1/퐴퐵 − 1/퐴퐶 3. Given three points {퐴, 퐵, 퐶} on line 휀, an arbitrary number 푘, and taking three other points {퐴′, 퐵′, 퐶′} on 휀, such that

(퐵퐶, 퐴퐴′) = (퐶퐴, 퐵퐵′) = (퐴퐵, 퐶퐶′) = 푘, implies (퐵′퐶′, 퐴퐴′) = (퐶′퐴′, 퐵퐵′) = (퐴′퐵′, 퐶퐶′) = 푘.

푀퐵/퐴퐵 − 푀퐷/퐴퐷 4. For arbitrary 푀 ∈ 휀 ∶ (퐴퐵, 퐶퐷) = . 푀퐵/퐴퐵 − 푀퐶/퐴퐶 5. For two quadruples of line 휀 satisfying (퐴퐵, 퐶퐷) = (퐴′퐵′, 퐶′퐷′) ⇒

퐴퐵 ⋅ 퐶퐷 퐴퐶 ⋅ 퐷퐵 퐴퐷 ⋅ 퐵퐶 + + = 0, (14) 퐴′퐵′ 퐴′퐶′ 퐴′퐷′ 퐴′퐵′ ⋅ 퐶′퐷′ 퐴′퐶′ ⋅ 퐷′퐵′ 퐴′퐷′ ⋅ 퐵′퐶′ + + = 0, (15) 퐴퐵 퐴퐶 퐴퐷

Nr‑1 and nr‑2 follow using the definition (퐴퐵, 퐶퐷) = (퐴퐶/퐵퐶) ∶ (퐴퐷/퐵퐷). BIBLIOGRAPHY 15

Nr‑3 follows by noticing the relations

(퐶퐵, 퐴′퐴) = (퐶퐴, 퐵퐵′) = (퐶퐶′, 퐴퐵) = 푘, which follow from the assumed equalities and the equalities in section 1. nr‑4 follows from simple calculations using “Euler’s formula” for four points on a line

푃퐴 ⋅ 퐵퐶 + 푃퐵 ⋅ 퐶퐴 + 푃퐶 ⋅ 퐴퐵 = 0, by replacing 푃 with 푀 and considering it also for 퐶 = 퐷.

Bibliography

[Ber87] Marcel Berger. Geometry vols I, II. Springer Verlag, Heidelberg, 1987.

[Cha52] M. Chasles. Traite de Geometrie Superieure. Bachelier, Paris, 1852.

[Pap96] Georges Papelier. Exercices de Geometrie moderne. Editions Jacques Gabay, 1996.

[Sch79] Hans Schwerdtfeger. Geometry of complex numbers. Dover, New York, 1979.

[VY10] Oswald Veblen and John Young. Projective Geometry vol. I, II. Ginn and Company, New York, 1910.

Related topics 1. Barycentric coordinates 2. Homographic Relation 3. Projective line 4.

Any correction, suggestion or proposal from the reader, to improve/extend the exposition, is welcome and could be send by e‑mail to: [email protected]