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 A 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. C 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. A 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 E 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 A' B Gergone Ge of A'B'C' = Symmedian K of ABC I C μ B' A C' ′ ′ ′ Figure 7: Gergonne point 퐺푒 of 퐴 퐵 퐶 = symmedian point 퐾 of 퐴퐵퐶 △퐴′퐵′퐶′ (see figure 7).