Symmedian Point 2

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Symmedian

A file of the Geometrikon gallery by Paris Pamfilos

The reason is not to glorify “bit chasing”; a more fundamental issue is at stake here∶ Numerical subroutines should deliver results that satisfy simple, useful mathematical laws whenever possible. … Without any underlying symmetry properties, the job of proving interesting results becomes extremely unpleasant. The enjoyment of one’s tools is an essential ingredient of successful work.

Donald Knuth, Seminumerical Algorithms, section 4.2.2

Contents (Last update: 15‑05‑2021)

1 Symmedians and symmedian point 2

2 Antiparallels 2

3 Harmonic quadrangle 3

4 Second Brocard triangle 5

5 Gergonne point 6

6 First Lemoine circle 7 7 Adams’ circle 9 8 Conics through parallels 10

9 Characterization of the first Lemoine circle 10 10 Second Lemoine circle 12 11 Inscribed rectangles 12

12 Medial and Orthic triangle intersections 13

13 Vecten squares of the triangle 14

14 Artzt parabolas 15 1 Symmedians and symmedian point 2

1 Symmedians and symmedian point

Here we discuss two related concepts∶ the “symmedian line” and the “symmedian point” or “Lemoine point” of a triangle 휏 = 퐴퐵퐶. The symmedian line of 휏 from vertex 퐴 is the symmetric of its median from 퐴 w.r. to the bisector from 퐴. The symmedian point 퐾 is

A A

Y'' X'' X' K K Y' I G X Y B C B A A'' A' C s

Figure 1: Symmedian line from 퐴 and symmedian point 퐾 the intersection point of the symmedian lines from the three vertices (See Figure 1). Next theorem expresses a “characteristic property” of the points 푌 of a symmedian line.

Theorem 1. The distances of points 푌 of the symmedian line 퐴퐴푠 from 퐴, from the adjacent sides {퐴퐵, 퐴퐶}, are proportional to these sides∶

푌푌′ 퐴퐵 = . (1) 푌푌″ 퐴퐶 Proof. This follows from the symmetry w.r.t. the bisector 퐴퐴″ and the corresponding property of the points 푋 of the median 퐴퐴′, by which, due to the equality of the areas of triangles {퐴퐵퐴′, 퐴퐴′퐶}, the analogous ratio is inversely proportional to the adjacent sides∶

푋푋′ 퐴퐶 = . (2) 푋푋″ 퐴퐵

Projecting the intersection point 퐾 of two of the symmedians on the sides of the tri‑ angles and applying the theorem, we see that the third symmedian passes also through that point, thus∶

Theorem 2. The three symmedians of the triangle pass through a common point 퐾 called the symmedian point of the triangle.

2 Antiparallels

Closely related to the concept of “symmedian” is the concept of “antiparallels” to a side of the triangle. An antiparallel 휀 = 퐵′퐶′ to side 퐵퐶 of the triangle is the line defined by any circle 휆 passing through 퐵, 퐶 and intersecting the sides {퐴퐵, 퐴퐶} a second time at the points {퐶′, 퐵′} (See Figure 2). There are two prominent antiparallels to 퐵퐶 ∶ 1. The tangent 휀0 to the circumcircle at 퐴. 2. The side 퐵″퐶″ of the “orthic triangle”, defined by the feet of the altitudes of the triangle. 3 Harmonic quadrangle 3

A ε

B'' ε 0 B' C'' κ

C' λ

B A'' C

Figure 2: Antiparallels 휀 to the side 퐵퐶

′ ′ The tangent 휀0 is the limiting position for which the points {퐵 , 퐶 } coincide with 퐴 and the circle 휆 coincides with the circumcircle 휅 of the triangle. The line 퐵″퐶″ is antiparallel because the quadrangle 퐵퐶퐵″퐶″ is “cyclic”. Theorem 3. All triangles 퐴퐵′퐶′ are similar to 퐴퐵퐶. This results immediately from the corresponding angle equalities of the triangles∶ 퐶̂′ = 퐶̂ and 퐵̂′ = 퐵.̂ (3) ′ ′ The symmetry on the bisector from 퐴 maps triangle 퐴퐵 퐶 to a triangle 퐴퐵1퐶1 with side ′ ′ 퐵1퐶1 parallel to 퐵퐶 hence the median line from 퐴 of 퐴퐵 퐶 maps to the median line of 퐴퐵퐶. This proves the following∶ Theorem 4. The symmedian of 퐴퐵퐶 coincides with the median line of the triangles {퐴퐵′퐶′}.

3 Harmonic quadrangle

Theorem 5. The tangents at {퐵, 퐶} to the circumcircle of triangle 퐴퐵퐶 intersect at a point 퐷 on the symmedian 퐴퐴푠. Proof. This follows from the characteristic property of the points of the symmedian line expressed by equation 1. Calculating the distances of 퐷 from the sides and using the sines‑rule we get (See Figure 3)∶ 퐷퐷′ 퐷퐵 sin(퐶)̂ sin(퐶)̂ 퐴퐵 = = = . 퐷퐷″ 퐷퐶 sin(퐵)̂ sin(퐵)̂ 퐴퐶

Theorem 6. The intersection 퐴푡 of the tangent to the circumcircle at 퐴 with 퐵퐶 is the “har‑ ′ ′ monic conjugate” of 퐴푠 w.r. to {퐵, 퐶} and the ratios 푋퐵 /푋퐶 defined by the sides {퐴퐵, 퐴퐶} and the symmedian line from 퐴 on every parallel 퐵′푋퐶′ to the side 퐵퐶 are equal to the ratio sin(퐶)̂ 2/ sin(퐵)̂ 2 = 푐2/푏2. 3 Harmonic quadrangle 4

Χ κ Β' C' K D''

A s A B C t E

D D'

Figure 3: Tangents intersecting on the symmedian 퐴퐴푠

Proof. This because the triangles {퐴퐴푡퐵, 퐶퐴푡퐴} are similar, implying (See Figure 3)∶

퐴 퐶 퐴 퐶 ⋅ 퐴 퐵 퐴 퐴2 sin(퐵)̂ 2 퐴 퐶 푡 = 푡 푡 = 푡 = = 푠 . 2 2 2 퐴푡퐵 퐴푡퐵 퐴푡퐵 sin(퐶)̂ 퐴푠퐵

It follows that the pencil of lines 퐴(퐵, 퐶, 퐴푠, 퐴푡) is harmonic, hence it defines on the circumcircle a “harmonic quadrangle”, i.e. a circumscribed quadrangle for which the “cross ratio” (퐵퐶퐴퐸) = (퐵퐶퐴푡퐴푠) = −1. This kind of quadrangles is also characterized by the property of being cyclic and having equal products of opposite sides.

Theorem 7. The intersection 퐸 of the symmedian from 퐴 with the circumcircle of 퐴퐵퐶 defines a harmonic quadrangle 퐴퐵퐸퐶.

κ λ

C' O D

E P

A B

Figure 4: Property of the symmedian related to the harmonic quadrangle 퐴퐵퐸퐶

Theorem 8. The circle 휆 through {퐵, 퐶} and the intersection 퐷 of the tangents at 퐵, 퐶 to the circumcircle 휅(푂) of the triangle 퐴퐵퐶 passes through the center 푂 and intersects the symmedian 4 Second Brocard triangle 5

퐴퐷 a second time at 푃, which defines triangles {퐴푃퐶 , 퐵푃퐶} simmilar to 퐵퐸퐶, so that the symmedian 푃퐸 bisects the angle 퐵푃퐶.̂ In addition holds the relation 푃퐴2 = 푃퐵 ⋅ 푃퐶. Proof. The angles {푂푃퐷, 푂퐶퐷} are right showing that 휆 passes through 푂 (see figure 4). A similar angle chasing shows that the angles of the triangles at 푃, 퐸 are equal to the supplement of 퐴̂. Note here that, since 퐷 is the middle of the arc 퐵퐶 on 휆 the symmedian 퐴퐷 bisects the angle 퐵푃퐶̂ . Also 퐸퐶퐵̂ = 퐵퐴푃̂ completes the proof that △퐴퐵푃 ∼ △퐶퐵퐸 . Analogously 퐸퐵푃̂ = 푃퐴퐶̂ completes the proof that △퐵퐸퐶 ∼ △퐴푃퐶 and shows that 푃퐸 is a bisector of the angle 퐵푃퐶̂ . For the proof of the last claim see that the parallel 퐶퐶′ to the symmedian 퐴퐷 is parallel to the bisector of 퐵푃퐶 hence 푃퐶 = 푃퐶′ and line 푃퐵 passes through 퐶′, hence 푃퐵⋅푃퐶=푃퐵⋅푃퐶′ = 푃퐴2. Figure 4 shows also a so‑called “Artzt parabola” of the triangle 퐴퐵퐶, which is the parabola passing through {퐵, 퐶} and being there tangent to the sides {퐴퐵, 퐴퐶}. Its focus is point 푃 (see section 14).

4 Second Brocard triangle

The “second Brocard triangle” of the triangle 퐴퐵퐶 is defined by its vertices which are the projections {퐴2, 퐵2, 퐶2} of the circumcenter 푂 of △퐴퐵퐶 on respective symmedians.

κ B 2 K λ

C 2

A O 2 B C

Figure 5: The second Brocard triangle 퐴2퐵2퐶2 of △퐴퐵퐶

Theorem 9. The circumcircle 휆 of the second Brocard triangle has the symmedian point 퐾 and the circumcenter 푂 of △퐴퐵퐶 as diametral points.

Proof. This follows directly from the definition of the vertices {퐴2, 퐵2, 퐶2} as projections of 푂 on respective symmedians, implying that each of them is viewing the segment 푂퐾 under a right angle.

The circle with diameter 푂퐾 is called “Brocard circle” of the triangle. We’ll see in theorem 25 that {퐴2, 푂} are inverses w.r.t. the Apollonian circle 휆퐴 through 퐴. This implies that the Brocard circle is orthogonal to 휆퐴. Analogously it is seen that this circle is orthogonal to all three Apollonian circles {휆퐴, 휆퐵, 휆퐶} of △퐴퐵퐶. This implies that the circle 휆, like the circumcircle 휅 belongs to the pencil of circles which are orthogonal to the Apollonian circles, called “Schoute pencil” of the triangle 퐴퐵퐶 (see file Pedal triangles). Theorem 10. The “cyclocevian” triangle 퐸퐹퐺 of the vertices of the second Brocard triangle is congruent and inversely oriented to the triangle 퐴퐵퐶. 5 Gergonne point 6

A F

κ G K O A 2

B C

Figure 6: The cyclocevian of 퐴2

Proof. “Cyclocevian” of a point 푃 w.r.t. to the triangle 퐴퐵퐶 is called the triangle formed by the second intersections of the lines {푃퐴, 푃퐵, 푃퐶} and the circumcircle 휅. We prove the theorem for 퐴2. In the course of proof of theorem 8 we have seen that drawing parallels to the symmedian 퐴퐾 from {퐵, 퐶} we get respectively their second intersections {퐺, 퐹} with the circumcircle 휅 , which are respectively collinear with {퐴2퐶, 퐴2퐵} (see figure 6). This implies easily that the two triangles {퐴퐵퐶, 퐸퐺퐹} are each the reflection of the other relative to the line 푂퐴2.

5 Gergonne point

′ ′ ′ ′ ′ ′ The “Gergonne point” 퐺푒 of △퐴 퐵 퐶 is the intersection point of the lines {퐴퐴 , 퐵퐵 , 퐶퐶 } joining the vertices with the contact points of the opposite sides with the “incircle” 휇(퐼) of

Gergone Ge of A'B'C' = Symmedian K of ABC

I C

B' A C'

′ ′ ′ Figure 7: Gergonne point 퐺푒 of 퐴 퐵 퐶 = symmedian point 퐾 of 퐴퐵퐶

△퐴′퐵′퐶′ (see figure 7). A direct consequence of theorem 5 is the following, which proves also the existence of this point. Theorem 11. The three lines {퐴퐴′, 퐵퐵′, 퐶퐶′} are symmedians of the triangle 퐴퐵퐶 and the Ger‑ ′ ′ ′ gonne point 퐺푒 of 퐴 퐵 퐶 coincides with the symmedian point 퐾 of 퐴퐵퐶. The two triangles {퐴퐵퐶, 퐴′퐵′퐶′} being “point perspective” w.r. to point 퐾, are also, according to “Desargues’ theorem” (see file Desargues’ theorem), “line perspective” i.e. cor‑ responding sides meet on a line 휀. This is, per definition the “trilinear polar” of 퐾, called “Lemoine line” of the triangle (see figure 8). 6 First Lemoine circle 7

C''

A''

ε B'

A C K B''

O κ A'

C' B

Figure 8: Lemoine line 휀 of triangle 퐴퐵퐶

From theorem 6 follows the following∶

Theorem 12. Points {퐴″, 퐵″, 퐶″} are the poles of lines, respectively {퐴퐴′, 퐵퐵′, 퐶퐶′} relative to the circumcircle 휅 of 퐴퐵퐶.

By the duality of the polarity, this implies that 퐾 is the pole of the trilinear polar 휀. Hence line 푂퐾 is orthogonal to the trilinear polar. This is the so‑called “Brocard axis” of the triangle (see file Brocard). By the fundamental property of “Apollonian circles” (see file Apollonian), by which these circles are orthogonal to 휅, follows the next property∶

Theorem 13. The points {퐴″, 퐵″, 퐶″} are the centers of the Apollonian circles, passing corre‑ spondingly through the vertices {퐴, 퐵, 퐶} of the triangle 퐴퐵퐶.

6 First Lemoine circle

The “first Lemoine circle” of the triangle 퐴퐵퐶 results by drawing parallels to the sides of the triangle from the symmedian point 퐾.

C 1 C' B'

B 1

A K' A 2 3 K

C B C D M B 2 3

Figure 9: First Lemoine circle of the triangle 퐴퐵퐶 6 First Lemoine circle 8

Theorem 14. The parallels to the sides of the triangle from the symmedian point intersect the sides in six concyclic points.

The proof results from the fact that the three segments {퐵1퐶1, 퐶2퐴2, 퐴3퐵3} joining the intersection points (see figure 9) are antiparallel respectively to the sides {퐵퐶, 퐶퐴, 퐴퐵}. To ′ see this for 퐵1퐶1 reflect 퐾 on the bisector 퐴퐷 to the point 퐾 , which is on the median ′ ′ ′ 퐴푀. Then, the whole parallelogram 퐴퐵1퐾퐶1 is reflected to the congruent toit 퐴퐵 퐾 퐶 , ′ ′ which has the diagonal 퐵 퐶 parallel to 퐵퐶, thus proving that 퐵1퐶1 is antiparallel to 퐵퐶. Analogously is proved the same property for the segments {퐶2퐴2, 퐴3퐵3}. This implies that the quadrangles 퐵1퐶1퐴3퐴2, 퐶2퐴2퐵1퐵3, 퐴3퐶1퐶2퐵3 (4) are cyclic. From this follows that 퐶1̂퐵1퐴 = 퐵퐴̂2퐶2 = 퐶.̂ This, in turn, implies that the trapezium 퐶1퐵1퐴2퐶2 is isosceles and inscribed in the circumcircle of 퐶1퐵1퐴2퐴3 . From this follows that the three quadrangles (4) have pairwise three common points, conse‑ quently their circumcircles coincide and define the so called “first Lemoine circle” of the triangle.

′ ′ ′ The lines {퐵1퐶1, 퐶2퐴2, 퐴3퐵3} define also another triangle 퐴 퐵 퐶 with remarkable prop‑

A C'

B 1

C 1 B'

K A A 2 3

A s B C C B 2 3

Figure 10: Additional properties related to the Lemoine circle erties (see figure 10). Theorem 15. Related to the Lemoine circle of the triangle 퐴퐵퐶 and the triangle 퐴′퐵′퐶′ are the following properties.

1. The segments {퐵1퐶1, 퐶2퐴2, 퐴3퐵3} are equal.

2. The triangles {퐴퐵1퐶1, 퐵퐶2퐴2, 퐶퐴3퐵3} are similar to 퐴퐵퐶. ′ ′ ′ 3. The triangles {퐴 퐵3퐶2, 퐵 퐶1퐴3, 퐶 퐴2퐵1} are isosceli. 4. The incircle 휆 of 퐴′퐵′퐶′ is concentric to the Lemoine circle 휅 of 퐴퐵퐶 . 7 Adams’ circle 9

5. The symmedians {퐴퐾, 퐵퐾, 퐶퐾} pass respectively through {퐴′, 퐵′, 퐶′}. ′ ′ ′ 6. The Gergonne point 퐺푒 of 퐴 퐵 퐶 coincides with the symmedian point 퐾 of 퐴퐵퐶. nr‑1, nr‑2 and nr‑3 are immediate consequences of theorem 14. nr‑4 follows from the equality of segments in nr‑1, since these are equal chords of the Lemoine circle. nr‑5 is proved by showing that for the symmedian line 퐴퐾퐴푠 the ratios 퐾퐴2/퐾퐴3 and 퐴푠퐶2/퐴푠퐵3 are equal∶ 퐴 퐶 퐴 퐵 − 퐶 퐵 푡 ⋅ 퐾퐴 − 퐾퐴 퐾퐴 푠 2 = 푠 2 = 2 2 = 2 . 퐴푠퐵3 퐴푠퐶 − 퐵3퐶 푡 ⋅ 퐾퐴3 − 퐾퐴3 퐾퐴3 Nr‑6 is an immediate consequence of the previous properties. Notice that the first Lemoine circle is a special case of the socalled “Tucker circles” of the triangle (see file Tucker circles).

7 Adams’ circle

The first Lemoine circle of 퐴퐵퐶 seen as a construct relative to the triangle 퐴′퐵′퐶′ is the so‑called “Adams’ circle” of the triangle 퐴′퐵′퐶′ , reflecting a property, usually formulated as follows ([Hon95, p.63])∶

A C'

B 1 A'' λ C 1 κ B'

K=G' A e A 2 3

B'' C'' A s B C C B 2 3

Figure 11: The Adams’ circle of the triangle 퐴′퐵′퐶′

″ ″ ″ ′ Theorem 16. The parallels to the sides of the cevian triangle 퐴 퐵 퐶 of the Gergonne point 퐺푒 ′ ′ ′ ′ ′ ′ ′ of the triangle 퐴 퐵 퐶 from point 퐺푒 intersect the sides of 퐴 퐵 퐶 in 6 points lying on a circle 휆 concentric to the incircle 휅 of 퐴′퐵′퐶′. The proof follows from the discussion in section 7 and the obvious fact that 퐴″퐵″퐶″ is ′ homothetic to 퐴퐵퐶 relative to the homothety with center 퐾 = 퐺푒 and ratio 1/2. 8 Conics through parallels 10

8 Conics through parallels

This and the next section give another aspect of the first Lemoine circle. We start witha point 퐷 not lying on the side‑lines of the triangle 퐴퐵퐶. Of central importance for this aspect is the following property. Theorem 17. The parallels to the sides of a triangle 퐴퐵퐶 passing through a point 퐷 not lying on the side‑lines of the triangle define respectively on the non‑parallel sides six points lying ona conic.

Z' D Y

B X X' C

Figure 12: The conic defined by parallels to the sides from 퐷

Proof. The proof is a trivial verification of “Carnot’s theorem” ([Yiu13, p.117]), by which the intersection points {푋, 푋′, …} are on a conic if and only if the product of ratios 푋퐵 푋′퐵 푌퐶 푌′퐶 푍퐴 푍′퐴 ⋅ ⋅ ⋅ ⋅ ⋅ = 1. 푋퐶 푋′퐶 푌퐴 푌′퐴 푍퐵 푍′퐵 In our case this product is easily seen to be equal to 푎2 푏2 푐2 ⋅ ⋅ = 1, 푏2 푐2 푎2 where {푎, 푏, 푐} the side lengths of the triangle. Hence the points are indeed on a conic.

Notice that this property can be generalized in the following sense (See Figure 13). Theorem 18. Given a line 휀 intersecting the sides of the triangle 퐴퐵퐶 at the points {퐴′, 퐵′, 퐶′}, consider the intersections {푋, 푋′, 푌, 푌′, 푍, 푍′} of the sides of the triangle with the lines through {퐴′, 퐵′, 퐶′} and an arbitrary point 퐷. Then the six points lie on a conic. The proof of this is reduced to the one of the previous theorem by means of a “projec‑ tivity” leaving the points {퐴, 퐵, 퐶} fixed and sending the line 휀 at the line at infinity.

9 Characterization of the first Lemoine circle

Next theorem characterizes the “symmedian point” of the triangle as the one for which the previous construction of conics delivers a circle. Here the symmedian point 퐾 of the triangle 퐴퐵퐶 is identified by its property to have distances from the sides proportional to these. Thus its “trilinear coordinates” are described by 퐾 ≅ (푎 ∶ 푏 ∶ 푐). 9 Characterization of the first Lemoine circle 11

Y' ε Z Y

Z' A' BCX X'

Figure 13: Conic defined by a point and a line

Theorem 19. With the notation and conventions of the preceding section the conic is a circle if and only if the point 퐷 coincides with the symmedian point of the triangle 퐴퐵퐶.

A Y'

x Z B Y Z' D

B X X' C

Figure 14: The conic is a circle

Proof. If the conic is a circle (See Figure 14), then the three trapezia {푋′푌푌′푍, …} are equi‑ lateral and the three segments {푋′푌, 푌′푍, 푍′푋} have equal lengths. By the equality of the inscribed angles seen in the figure follows that the distance 푥퐵 of 퐷 from 퐴퐶 is ′ 푥퐵 = 푍푌 sin(퐵)̂ . This implies immediately

(푥퐴 ∶ 푥퐵 ∶ 푥퐶) = (sin(퐴)̂ ∶ sin(퐵)̂ ∶ sin(퐶)̂ = (푎 ∶ 푏 ∶ 푐), thereby proving the necessity part of the theorem. For the sufficiency consider the triangle 퐷푍푌′. By assumption its altitudes satisfy

푏 푥 푍푌′ sin(푍)̂ sin(푍)̂ 퐷푌′ = 퐵 = = = . 푐 푥퐶 푍푌′ sin(푌̂′) sin(푌̂′) 퐷푍

Thus, the triangles {퐴퐵퐶, 퐷푍푌′}, having the angles 퐴̂ = ̂퐷 and the containing them sides proportional, are similar and the angles 푍̂ = 퐵.̂ Analogous argument shows that the angle 10 Second Lemoine circle 12

푌푋̂′푍 is equal to 퐵̂. This implies that 푋′푌푌′푍 is an isosceles trapezium inscribed in a circle passing through 푋. A similar argument shows the analogous property for the trapezia 푌′푍푍′푋 and 푍′푋푋′푌. This implies that all six points {푋, 푋′, 푌, 푌′, 푍, 푍′} are on the same circle.

10 Second Lemoine circle

The “second Lemoine circle” of the triangle 퐴퐵퐶 is a circle with center at the symmedian point 퐾 , whose existence follows from next theorem

B C 1 1 A 2

Κ κ

A 1

Β C B C 2 2

Figure 15: Second Lemoine circle of the triangle

Theorem 20. (See Figure 15). On the antiparallels to the sides of the triangle 퐴퐵퐶, which are drawn from the symmedian point 퐾 , the other sides define respectively equal segments {퐾퐴1 = 퐾퐴2, …}, which define diameters of a circle 휅 with center 퐾.

The quadrangles 퐴1퐵1퐴2퐵2, 퐵1퐶1퐵2퐶2, 퐶1퐴2퐶2퐴1 (5) are rectangles. For 퐴1퐵1퐴2퐵2 , using the fact that the quadrangles

퐴1퐴2퐶퐵, 퐵1퐵2퐶퐴, 퐶1퐶2퐵퐴 are cyclic, this follows from the equality of the angles noticed in figure 15. Analogously is proved the property for the other quadrangles in (5). The rectangles (5) have pairwise one common diagonal, and this proves the theorem.

11 Inscribed rectangles

In this section we consider rectangles inscribed in a given triangle, with one side coinci‑ dent with a side of the triangle. Next theorem relates these rectangles with the symmedian point of the triangle (see figure 16).

Theorem 21. The centers 푀 of the rectangles, inscribed in the triangle 퐴퐵퐶 and having one side on the line 퐵퐶, are contained in a line 휀, which passes through the middle 푁 of 퐵퐶, the middle 퐿 of the altitude 퐴푌 and the symmedian point 퐾 of the triangle. 12 Medial and Orthic triangle intersections 13

Α ε

L D P Η

Κ Μ

ΒΖCΕY Ι Ν

Figure 16: Inscribed rectangles

In fact, the centers 푀 of the rectangles are the middles of the segments 푃퐼 joining the middles of opposite parallels {퐷퐻, 푍퐸}. Since all triangles {푃퐼푁} are similar, the claim that 푀 describes a line passing through {푁, 퐿} is clear. That this line passes also through 퐾 follows from the proof of theorem 20, where we saw that there is such a rectangle with center at 퐾 .

B'' C'' L N 1 N 3 2

K L L 2 3

B C A'' N 1

Figure 17: Alternative definition of the symmedian point

Theorem 22. The symmedian point 퐾 of the triangle is the common point of the lines joining the middles {퐿푖} of the altitudes with the middles {푁푖} of the opposite sides of the triangle 퐴퐵퐶. This is a direct corollary of theorem 21. Notice that this point 퐾 is the “triangle center” 푋(69) w.r. to the medial triangle 푁1푁2푁3 ([Kim18]).

12 Medial and Orthic triangle intersections

Consider the “medial triangle” 퐴″퐵″퐶″ with vertices the middles of the sides of triangle 퐴퐵퐶. Each symmedian line of 퐴퐵퐶 intersects the sides of the medial triangle, which are adjacent to the opposite vertex of the corresponding parallelogram at two points. For ex‑ ″ ″ ″ ″ ample the symmedian 퐵퐾 intersects the sides {퐵 퐴 , 퐵 퐶 } at the points {푉2, 푈2}. Anal‑ ogously are defined the points {푉1, 푈1} and {푉3, 푈3} (see figure 18). Next theorem lists some properties of these points. Theorem 23. Let 퐴′퐵′퐶′ be the “orthic” triangle of 퐴퐵퐶, with vertices the feet of the altitudes. Then the following are valid properties. 13 Vecten squares of the triangle 14

1. The sides of the orthic at {퐴′, 퐵′, 퐶′} pass respectively through the points {(푈1, 푉1), (푈2, 푉2), (푈3, 푉3)}. ′ ″ ′ ″ 2. The triangles {퐴 퐴 푉1, … , 퐴 퐴 푈1, …} are all similar to 퐴퐵퐶. ′ ′ To show nr‑1 in a case, for example, to show that 퐴 퐶 passes through 푉1 , consider for ′ ′ ″ ″ ′ ″ the moment 푉1 as the intersection of lines {퐴 퐶 , 퐴 퐵 }. Measuring the angles of 퐴 퐴 푉1 ′ ″ we see that this triangle is similar to 퐴퐵퐶 and the quadrangle 퐴 푉1퐶퐵 is cyclic. All the angles at {퐴′, 퐴″, 퐵″} can be measured using standard properties of the orthic triangle and ″ show that 퐵 퐶푉1 is similar to 퐴퐵퐶.

V B' 2

V3 C'' B'' U 2 κ K C' U3

U1 Q P

C B A' A''

Figure 18: Intersections of sides of the medial and the orthic

Having that, project 푉1 on the sides of 퐴퐵퐶 at points {푆, 푄}. Then we have∶

′ ″ 푉1푆 푃퐶 퐶푃 퐶퐵 퐴퐵 = = = ″ = , 푉1푄 푉1푄 푉1푄 푉1퐵 퐴퐶 proving that 푉1 satisfies the characteristic property of the points of the symmedian, hence it is on the symmedian 퐴퐾. Analogously are proved the other cases of this claim. Nr‑2 results, as alluded to before, by measuring the angles of these triangles. Figure 18 ′ ″ gives a hint for this in the case of the triangle 퐴 퐴 푉1.

13 Vecten squares of the triangle

On the sides of triangle 휏 = 퐴퐵퐶 erect squares. These are the “Vecten squares” of the triangle. The triangle of the opposite sides of the squares 휏′ = 퐴′퐵′퐶′ has its sides parallel to those of 휏, hence is similar to it. (see figure 19). 14 Artzt parabolas 15

A''

A' A

K B C

B' C'

Figure 19: Vecten squares of the triangle

Theorem 24. The similarity center of {휏, 휏′} is the symmedian point 퐾 of 휏, which is also the symmedian point of 휏′.

The proof is immediately seen in the figure∶

퐴퐴′ 퐴퐵 퐴′퐵′ = = , 퐴퐴″ 퐴퐶 퐴′퐶′ showing that 퐴 is on the symmedian line of 휏′ and 휏. The same result holds for the configuration created by erecting squares on the sides, each lying on the same side with the opposite vertex (see figure 20). The only difference in this case is that the two triangles 퐴퐵퐶 and 퐴′퐵′퐶′ are related by an “anti‑homothety” with center the symmedian point 퐾 of both triangles.

A' C' A

B C

Figure 20: Inner Vecten squares of the triangle

14 Artzt parabolas

The following properties relate the symmedian 퐴퐾 to the Apollonian circle 휆퐴(푂퐴), which is the locus of the points {푋 ∶ 푋퐵/푋퐶 = 푏/푐} and lead also to a characterization 14 Artzt parabolas 16

K O A 2 O B A λ C Α

Figure 21: The Artzt parabolas of △퐴퐵퐶

of the projection 퐴2 of the circumcenter on the symmedian 퐴퐾 as the focus of the “A‑ Artzt parabola” of the triangle 퐴퐵퐶 . By its definition, this parabola is tangent to the sides {퐴퐵, 퐴퐶} at the points {퐵, 퐶.} Analogously are defined the B‑Artzt parabola tangent to the sides {퐵퐶, 퐵퐴} and the C‑Artzt parabola tangent to {퐶퐴, 퐶퐵} (see figure 21). Notice that point 퐴2 is a vertex of the so called “second Brocard triangle” ([Cou80, p.279]). Its isogo‑ nal conjugate 퐽 appearing in the next theorem is a vertex of the “fourth Brocard triangle” [Gib21]).

λ Α A

J κ

H K G O A Ο 2 Α B M C

E F

Figure 22: The projection 퐴2 of 푂 on the symmedian 퐴퐾

Theorem 25. Referring to figure 22, we denote by 휅(푂) the circumcircle of △퐴퐵퐶 and consider the points: the second intersection 퐴′ of the symmedian 퐴퐾 with 휅, the second intersection 퐹 of the median 퐴푀 with 휅 and the reflection 퐽 of 퐴′ in 퐵퐶. 1. 퐵퐴′퐹퐶 is a trapezium and 퐴′퐹 is parallel to 퐵퐶. ′ ′ ′ 2. Point 퐴 is on the Apollonian circle 휆퐴(푂퐴) , satisfying 퐴 퐵/퐴 퐶 = 푏/푐. ′ 3. Point 퐴2 is the middle of 퐴퐴 , line 푂퐴2 passes through 푂퐴.

4. Point 퐴2 is the inverse of 푂 w.r.t. to 휆퐴.

5. Point 퐽 is the second intersection of the Apollonian circle 휆퐴 with the median 퐴푀. 14 Artzt parabolas 17

6. 퐽퐵퐹퐶 is a parallelogram and point 퐽 is the isogonal conjugate of 퐴2 . 7. Line 퐻퐽 is orthogonal to the median 퐴푀, where 퐻 is the orthocenter of △퐴퐵퐶.

Proof. Nr‑1 follows from the equality of the angles 퐵퐴퐴̂ ′ = 퐹퐴퐶̂ . Nr‑2: By nr‑1 and the similar triangles {푀퐹퐶 ∼ 푀퐵퐴 , 푀퐹퐵 ∼ 푀퐶퐴} we have

퐴′퐵 퐹퐶 퐹퐶 푀퐵 퐴퐵 퐴푀 퐴퐵 = = ⋅ = ⋅ = . 퐴′퐶 퐹퐵 푀퐶 퐹퐵 퐴푀 퐴퐶 퐴퐶 Nrs 3‑4 are a direct consequence of nr‑2. Nr‑5 Follows from the fact that 퐴′푀퐹 is isosceles and 퐽퐴̂′퐹 is a right angle. This implies that 퐽 is on the median 퐴퐹. Obviously it is also on 휆퐴 since 퐵퐶 is a diameter of this circle. Nr‑6: By nr‑4 point 푀 is the middle of both segments {퐵퐶, 퐽퐹} implying that 퐽퐵퐹퐶 is a parallelogram. By theorem 8 and the preceding claim these triangles are similar:

′ 퐵퐴2퐴 ∼ 퐴퐴2퐶 ∼ 퐵퐴 퐶 ∼ 퐶퐹퐵 ∼ 퐵퐽퐶.

This implies 퐴̂2퐵퐴 = 퐽퐵퐶̂ ⇒ 퐽퐵퐴̂ = 퐴̂2퐵퐶. Analogously is seen that 퐴̂2퐶퐵 = 퐽퐶퐴.̂ Since we have also 퐴̂2퐴퐵 = 퐽퐴퐶̂ the claim is proved. nr‑7: This involves a computation in barycentrics (see file Barycentric coordinates) in which the points have corresponding coordinates

2 퐴(1, 0, 0) , 퐺(1, 1, 1) , 퐻(푆퐵푆퐶, 푆퐶푆퐴, 푆퐴푆퐵) , 퐽(푎 , 2푆퐴, 2푆퐴).

′ ′ ′ We test the orthogonality formula for lines 푆퐴푝푝 + 푆퐵푞푞 + 푆퐶푟푟 = 0, where (푝, 푞, 푟) and (푝′, 푞′, 푟′) are the points at infinity (their directions) of the lines {퐴퐺, 퐻퐽} given by the respective triple products {(퐴 × 퐺) × 퐺 , (퐻 × 퐽) × 퐺}. The result is the expression 2 2(푆퐵 + 푆퐶 − 푎 )(푆퐵푆퐶 + 푆퐶푆퐴 + 푆퐴푆퐵) = 0, according to the identity for the “Conway sym‑ 2 bols” 푆퐵 + 푆퐶 = 푎 (see file Conway symbols).

A ε

O A* A 2

B J C

∗ ∗ Figure 23: Generating the Artzt parabola by varying triangle 퐵 퐴2퐴 ∼ 퐵퐴2퐴 BIBLIOGRAPHY 18

Next theorem complements some properties discussed in theorem 8. Point 퐷 is de‑ fined as the second intersection of the parallel 퐶퐷 to the symmedian 퐴퐴′ with the cir‑ cumcircle of triangle 퐴퐵퐶. In that theorem was shown that the projection 퐴2 of the cir‑ cumcenter on the symmedian 퐴퐴′, which is a vertex of the second Brocard triangle, was a point of the line 퐵퐷 (see figure 23).

Theorem 26. Referring to figure 23, the following are valid properties: ′ 1. Triangles 퐴2퐵퐴 and 퐴퐴2퐷 are similar to 퐴퐵퐶. ′ 2. Triangle 퐷퐴2퐶 is isosceles and triangles 퐴퐷퐴2 and 퐴 퐶퐴2 are equal. ′ ′ 3. Triangles 퐴2퐵퐴 and 퐴2퐴 퐶 are also similar. ∗ ∗ ∗ ∗ 4. Varying triangle 퐵퐴2퐴 to a similar 퐵 퐴2퐴 so that 퐵 moves on 퐴퐵 makes vertex 퐴 move on line 퐴퐶 and the line 휀 = 퐵∗퐴∗ is tangent to the parabola.

Nr‑1. In fact, △퐴퐵퐶 is similar to △퐴2퐴퐷. Later has 퐴퐷퐴̂ 2 = 퐴퐶퐵̂ and 퐴̂2퐴퐷 = 퐴퐵퐶,̂ ′ ′ ′ because 퐴퐷퐶퐴 is equilateral trapezium and 퐴̂2퐴퐷 = 퐴̂2퐴 퐶 = 퐴퐵퐶.̂ Obviously △퐴2퐵퐴 is similar to 퐴2퐴퐷. ′ Nr‑2 is obvious, since the triangles {퐴2퐴퐷 , 퐴2퐴 퐶} are each the reflection of the other in line 푂퐴2. Nr‑3 is a consequence of nr‑2. Nr‑4 is a general property of parabolas related to a triangle like 퐴퐵퐶 , which has two sides tangent to the parabola and the third side joins the contact points. In such a case it is well known ([Cha65, p.52]) that a line 퐴∗퐵∗ intersecting the tangents at points such that 퐴퐵∗/퐵∗퐵 = 퐶퐴∗/퐴∗퐴 is tangent to the parabola (see file Triangles tangent to parabolas). ∗ ∗ ∗ ∗ ∗ ∗ The relation 퐴퐵 /퐵 퐵 = 퐶퐴 /퐴 퐴 follows from the similar triangles {퐴퐵 퐴2 ∼ 퐶퐴 퐴2}.

Bibliography

[Cha65] M. Chasles. Traite de Sections Coniques. Gauthier‑Villars, Paris, 1865.

[Cou80] Nathan Altshiller Court. College Geometry. Dover Publications Inc., New York, 1980.

[Gib21] B. Gibert. Brocard triangles. https://bernard-gibert.pagesperso-orange. fr/gloss/brocardtriangles.html, 2021.

[Hon95] Ross Honsberger. Episodes in Nineteenth and Twentieth Century Euclidean Geome‑ try. The Mathematical Association of America, New Library, Washington, 1995.

[Kim18] Klark Kimberling. Encyclopedia of Triangle Centers. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html, 2018.

[Yiu13] Paul Yiu. Introduction to the Geometry of the Triangle. http://math.fau.edu/Yiu/ Geometry.html, 2013. BIBLIOGRAPHY 19

Related topics 1. Barycentric coordinates 2. Brocard 3. Cross Ratio 4. Desargues’ theorem 5. Pedal triangles 6. Projective line 7. The quadratic equation in the plane

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]