HYACINTHOS 6293

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* SHARYGIN POINTS REPORT *
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Prof. Clark Kimberling has asked me to record the theory of the
Sharygin points for referring to it in ETC. Here is it; for a more
complete (but very tedious) treatise of the first three Sharygin
points with full proofs and a few remarks on the next two, I refer to
my note [4]

"On the principal Sharygin points of a triangle",

which is avaliable at the Files Directory of Hyacinthos as
SHARYGIN.ZIP (as Zipped PS file).

The notations in this message are adjusted to those I used in [2] and
[4]. Although the first five Sharygin points will also get the
notions P, P', Q, T, T', generally the n-th Sharygin point will be
denoted by Sn.

CONTENTS

§1. Introduction
§2. Two triangles
§3. The first three Sharygin points S1, S2, S3
§4. The excentral triangle; Sharygin points S4, S5
§5. Centers of similtude; Jean-Pierre Ehrmann's
Sharygin points S6, S7
§6. The intouch triangle and S8
§7. The orthologic Sharygin point S9
§8. Desmic perspectors S10, S11, S12, S13, S14, S15
§9. Sharygin point list
§10. Historical remarks

§1. INTRODUCTION

We will treat two remarkable triangles: the triangle bounded by the
perpendicular bisectors of the internal angle bisectors of a triangle
ABC, and the triangle bounded by the perpendicular bisectors of the
external angle bisectors of triangle ABC. These two triangles and the
triangle ABC are three perspective triangles, having a common
perspectrix: the Lemoine axis of ABC. The mutual perspectors of the
three triangles will be called the first, second and third Sharygin
points of ABC (after a problem of Igor Sharygin - see §10). These
points will turn out to be X(256), X(291) and a new point in
Kimberling's list. Other concurrences will lead us to 12 other
Sharygin points - which have different significance and properties.

I express my thanks to Jean-Pierre Ehrmann for some properties of the
Sharygin points S3, S5, S6 and S7.

§2. TWO TRIANGLES

Let ABC be a triangle. The internal angle bisectors from A, B, C meet
BC, CA, AB at A', B', C'. The external angle bisectors from A, B, C
meet BC, CA, AB at A", B", C". Let X, X', Y, Y', Z, Z' be the
midpoints and x, x', y, y', z, z' the perpendicular bisectors of the
segments AA', AA", BB', BB", CC', CC" respectively.

Now define the points

D = y /\ z, E = z /\ x and F = x /\ y;
D' = y' /\ z', E' = z' /\ x' and F' = x' /\ y',

where the sign /\ means "intersection".

The triangles DEF and D'E'F' are the triangles enclosed by the
perpendicular bisectors of the internal and external angle bisectors
of ABC, respectively.

A little digression: We have some collinear triples of points: (A',
B', C"), (A', B", C'), (A", B', C'), (A", B", C"), (X, Y, Z'), (X,
Y', Z), (X', Y, Z) and (X', Y', Z') - proved in [4].

It is not hard to see ([4]) that x and x' pass through the midpoint
of A'A", y and y' through the midpoint of B'B", and z and z' through
the midpoint of C'C". But these midpoints lie on the Lemoine axis of
triangle ABC.

We note down the homogeneous trilinears of D and D':

/ -(aa - bc) 1 1 \
D ( ------------------ : ------- : ------- ) and
\ (bb + ca)(cc + ab) bb + ca cc + ab /

/ aa - bc 1 1 \
D' ( ------------------ : ------- : ------- )
\ (bb - ca)(cc - ab) bb - ca cc - ab /

proven in [4].

§3. THE FIRST THREE SHARYGIN POINTS S1, S2, S3

The triangles ABC, DEF and D'E'F' are mutually perspective, having a
common perspectrix - this is the Lemoine axis of triangle ABC. Now
consider their mutual perspectors:

The perspector of ABC and DEF is the 1ST SHARYGIN POINT of ABC,
denoted by P or S1, having homogenous trilinears

/ 1 1 1 \
P ( ------- : ------- : ------- ) = Kimberling's X(256).
\ aa + bc bb + ca cc + ab /

("aa" is an email shorthand for "a^2")

The perspector of ABC and D'E'F' is the 2ND SHARYGIN POINT of ABC,
denoted by P' or S2, having homogeneous trilinears

/ 1 1 1 \
P'( ------- : ------- : ------- ) = Kimberling's X(291).
\ aa - bc bb - ca cc - ab /

The perspector of DEF and D'E'F' is the 3RD SHARYGIN POINT of ABC,
denoted by Q or S3, having homogeneous trilinears

/ (aa - bc)(b^3 + c^3 - a^3 - abc) \
Q ( -------------------------------- : ... by cyclic permutation )
\ a /

This point is not yet in Kimberling's ETC list.

The three Sharygin points P, P', Q are collinear; I propose to call
the line PP'Q the SHARYGIN LINE of triangle ABC.

§4. THE EXCENTRAL TRIANGLE; SHARYGIN POINTS S4, S5

The excentral triangle OaObOc of ABC is homothetic to triangle DEF
(see [4]). The homothetic center (and perspector) of these two
triangles is the 4TH SHARYGIN POINT of ABC, denoted by T or S4, with
homogeneous trilinears

T ( -aa+bb+cc+ab+bc+ca
: -bb+cc+aa+ab+bc+ca
: -cc+aa+bb+ab+bc+ca ) = Kimberling's X(846).

The excentral triangle OaObOc is also perspective to triangle D'E'F'
(but not homothetic). The perspector T' is the 5TH SHARYGIN POINT. It
is not (yet) in Kimberling's ETC. The trilinears of T' are

T' ( a^4+(b+c)a^3-(2bb+3bc+2cc)aa+(b+c)(bb+cc)a-(b-c)^2(bb+bc+cc)
: ... by cyclic permutation )

§5. CENTERS OF SIMILTUDE; JEAN-PIERRE EHRMANN'S
SHARYGIN POINTS S6, S7

We conclude with some related Sharygin points not contained in [4].

Jean-Pierre Ehrmann noted that the sidelines of triangle D'E'F' are
parallel to the internal bisectors of triangle ABC, and therefore
orthogonal to the sides of the excentral triangle OaObOc. Hence,
triangles D'E'F' and OaObOc are directly similar. Their center of
similtude will be called the 6TH SHARYGIN POINT and has trilinears

( bb+cc-aa+a(b+c)-3bc : ... by cyclic permutation )
= Kimberling's X(1054).

The center of similtude of the triangles DEF and D'E'F' (which are
directly similar, too) is the 7TH SHARYGIN POINT with trilinears

( a(b^4+c^4-a^4+(b+c)a^3-(b^3+c^3)a-bbcc)
: ... by cyclic permutation ) = a point not in ETC.

These trilinears for S6 und S7 were found by Jean-Pierre Ehrmann, who
also gave the following corollaries:
The points S3 = Q and S7 are the common points of the circumcircles
of the triangles DEF and D'E'F'; the points S5 = T' and S6 are the
common points of the circumcircles of the triangles D'E'F' and OaObOc.

PROOF: This follows from the fact that (I cite Jean-Pierre)

"when two triangles are perspective and directly similar (but not
homothetic), the perspector and the center of similitude are the
common points of their circumcircles."

§6. THE INTOUCH TRIANGLE AND S8

Another interesting point is the homothetic center of the triangle
DEF and the intouch triangle of ABC. I call it the 8TH SHARYGIN POINT
of triangle ABC. It is not (yet) in ETC.

NOTES: 1. The intouch triangle is the triangle whose vertices are the
points of tangency of the incircle with the sides of triangle ABC. It
is also called the Gergonne triangle of ABC.
2. The triangle D'E'F' is not perspective to the intouch triangle of
ABC.

§7. THE ORTHOLOGIC SHARYGIN POINT S9

Another nice point, which is not yet in ETC, is the 9TH SHARYGIN
POINT, which I also call the ORTHOLOGIC SHARYGIN POINT. It is the
intersection of the perpendiculars from D to BC, from E to CA, and
from F to AB.

PROOF: The perpendiculars from D to BC, from E to CA, and from F to
AB concur, since the perpendiculars from A to EF, from B to FD, and
from C to DE concur (orthologic triangles!).

I have not managed to compute the trilinears of S8 and S9.

§8. DESMIC PERSPECTORS S10, S11, S12, S13, S14, S15

We conclude the study with some artificial concurrences; in fact, the
following concurrences follow from the desmic theory:

If P1P2P3 and Q1Q2Q3 are two arbitrary perspective triangles, and

R1 = P2Q3 /\ P3Q2;
R2 = P3Q1 /\ P1Q3;
R3 = P1Q2 /\ P2Q1,

then the triangle R1R2R3 is called the DESMIC MATE of triangles
P1P2P3 and Q1Q2Q3. Note that the "desmic mate" is a symmetric
relation: The triangle R1R2R3 is automatically the desmic mate of
triangles Q1Q2Q3 and P1P2P3. Now after the desmic theory, which goes
back to Floor van Lamoen, the triangles P1P2P3 and R1R2R3 are
perspective, and the triangles Q1Q2Q3 and R1R2R3 are perspective.

Applied to any two of the three mutually perspective triangles ABC,
DEF and D'E'F', we get the following new perspectors:

S10 perspector of ABC and the desmic mate of the triangles ABC and
DEF;
S11 perspector of DEF and the desmic mate of the triangles ABC and
DEF;
S12 perspector of ABC and the desmic mate of the triangles ABC and
D'E'F';
S13 perspector of D'E'F' and the desmic mate of the triangles ABC and
D'E'F';
S14 perspector of DEF and the desmic mate of the triangles DEF and
D'E'F';
S15 perspector of D'E'F' and the desmic mate of the triangles DEF and
D'E'F'.

I call these points the 10th till 15th Sharygin points of ABC.

The difficulty of the trilinear coordinates varies from point to
point, but the first four are quite simple:

S10 ( (aa + bc)(aa - bc)
: (bb + ca)(bb - ca)
: (cc + ab)(cc - ab) )
S11 ( (bb + ca)(cc + ab) - (aa + bc)(aa - bc)
: (cc + ab)(aa + bc) - (bb + ca)(bb - ca)
: (aa + bc)(bb + ca) - (cc + ab)(cc - ab) )
S12 ( (aa - bc)^2 : (bb - ca)^2 : (cc - ab)^2 )
S13 ( (bb - ca)(cc - ab) + (aa - bc)^2
: (cc - ab)(aa - bc) + (bb - ca)^2
: (aa - bc)(bb - ca) + (cc - ab)^2 )

As far as my (hastily made) macro was correct, none of the points S10-
S15 is in Kimberling's ETC.

§9. SHARYGIN POINT LIST

We sum up the Sharygin points in a little list:

S1 perspector of ABC and DEF = Kimberling's X(256)
S2 perspector of ABC and D'E'F' = Kimberling's X(291)
S3 perspector of DEF and D'E'F'
S4 perspector of DEF and OaObOc (and homothetic center)
= Kimberling's X(846)
S5 perspector of D'E'F' and OaObOc
S6 center of similtude of D'E'F' and OaObOc
= Kimberling's X(1054)
S7 center of similtude of D'E'F' and DEF
S8 perspector of DEF and intouch(=Gergonne) triangle
of ABC
S9 concurrence point of the perpendiculars from D to
BC, etc.
S10 perspector of ABC and the desmic mate of the
triangles ABC and DEF;
S11 perspector of DEF and the desmic mate of the
triangles ABC and DEF;
S12 perspector of ABC and the desmic mate of the
triangles ABC and D'E'F';
S13 perspector of D'E'F' and the desmic mate of the
triangles ABC and D'E'F';
S14 perspector of DEF and the desmic mate of the
triangles DEF and D'E'F';
S15 perspector of D'E'F' and the desmic mate of the
triangles DEF and D'E'F'.

It would be nice if somebody finds the trilinears of S8, S9, S14 or
S15.

§10. HISTORICAL REMARKS

Now I will briefly explain the reasons of my naming "Sharygin points".

In the article "Teoremy Chevy i Menelaja" by Igor Sharygin (who
actually is a member of Hyacinthos!) in the 11/1976 issue of the
Russian mathematics journal "Kvant" - these issues are avaliable at

http://kvant.mccme.ru/

-, I read the following problem:

To prove that the perpendicular bisectors of the angle bisectors of a
triangle intersect the corresponding sides on the triangle in three
collinear points.

This problem is (by Desargues' Theorem) equivalent to the fact that
the triangle enclosed by the perpendicular bisectors of the angle
bisectors is perspective to ABC. This was the origin of my theory of
the Sharygin points described here.

REFERENCES

[1] Darij Grinberg, "On Kimberling's X(256) and X(291) triangle
centers" in geometry-college,
http://mathforum.org/epigone/geometry-college/peeringglur

[2] Darij Grinberg, "On Kimberling's X(256) and X(291) triangle
centers ("Sharygin points") (Sequel)" and "On Kimberling's X(256) and
X(291) triangle centers ("Sharygin points") (Corrections)" in
geometry-college,
http://mathforum.org/epigone/geometry-college/peeringglur

[3] Jean-Pierre Ehrmann, personal correspondence.

[4] Darij Grinberg, "On the principal Sharygin points of a triangle",
SHARYGIN.ZIP in the Files directory of Hyacinthos.

Darij Grinberg