Ring of Tücker

Definition of the circles of Tücker

Definition 1: homothety

In a Homothety of center L, the Not of Lemoine, report/ratio K (K ≠ 1 and K ≠ 0), the triangle ABC has as an image has' B' It. The sides of the triangle has' B' It meet those of ABC in six points. These points are cocyclic and are located on a circle (T) known as of Tücker of the triangle ABC.

Properties
The mediums U, V, W of the segments, are located on the symédianes and form a homothetic triangle UVW of ABC in a homothety of center L.

The lines (MN), (PQ) and (RS) are antiparallel at the sides of the triangle and the segments which they determine are of the same length.

The center Ω of the circle (T) is the medium of the segment formed by the centers of the circles circumscribed with the triangles ABC and has' B' It.

Indications
The mediums U, V, W of the segments, are located on the symédianes, the segments are antiparallel with dimensions opposite.

See : medium of an antiparallel

The lines (MN, CB) are antiparallel with the right-hand sides (AB, CA): (AB, MN) = (CB, CA).
The lines (SR, (CA) are also antiparallel with the right-hand sides (BC, BA): (BA, SR) = (CA, CB).

One from of deduced that (BA, SR) = - (AB, MN).
Like (AB) // (NR) one a: (BA, SR) = - (NR, MN).
With the points of the hexagon (SM, SR) = (Nm, NR).
The points S, M, NR, R not being aligned, this equality of angles shows that they cocyclic, are located on a circle (T).

Of (BC)/(PS) and (MN) antiparallel with (BC) one of deduced that (PS) is antiparallel with (MN) compared to (ms) and (PN). (PS, PN) = (MS, MN). P, S, M, NR are cocyclic, P belongs to the circle containing S, M, NR: the circle (T). One shows just as (T) contains (Q).

Demonstrations : Left Yvonne and Rene - geometry of the triangle - Hermann 1997

Definition 2: Construction of an antiparallel

ABC is a triangle of Cercle circumscribed (Γ) of center O.

Starting from a point M of (AB) distinct to carry out the antiparallel line of (BC) compared to (AB, AC). It is the parallel to the tangent has some with (Γ), therefore perpendicular to (AO). It crosses (AC) in NR. the parallel to (AB) passing by NR crosses (BC) as a R. the circumscribed circle to triangle MNR recuts the sides of the triangle ABC out of P, Q and S.

We obtain a configuration of six points located on a circle of Tücker.

Properties
The parallel straight lines (AB) and (NR) cut the circle according to two equal cords, from where MN = SR.

(RS) antiparallel with (AC) compared to (BA, BC):

(RS, BA) = (RS, RN) because (BA)/(RN)
(RS, RN) = (ms, MN) = (AB, MN), inscribed angles of droites
(AB, MN) = (BC, AC) because the lines (MN), (BC) are antiparallel with right-hand sides (AB), (AC).

One thus has (RS, BA) = (BC, AC): lines (RS), (AC) are antiparallel with the right-hand sides (BA), (BC).

(MQ) parallel with (AC):

(MQ, AC) = (MQ, MN) + (MN, AC)
(MQ, MN) = (RQ, RN) = (RQ, AB), inscribed angles of droites
(MN, AC) = (AB, BC) because the lines (MN), (BC) are antiparallel with right-hand sides (AB), (AC)
(MQ, AC) = (RQ, AB) + (AB, BC) = (RQ, BC) = 0.

(MQ) // (AC). These parallels cut the circle according to two equal cords, from where MN = PQ and MN = PQ = SR.

Equality PQ = SR it results parallelism from (BC) and (SP).

A calculation of angles similar to the first calculation makes it possible to deduce that (PQ) is antiparallel with (AC) compared to (BA, BC).

Conclusions

The six points play of the similar roles. By each point one carries out two lines: one parallel with the tangent with (Γ) in one of the tops on the side which carries it, and the other parallel at the other side resulting from this top.

By any point on a side distinct from the tops passes two circles of Tücker obtained by considering the two tangents to (Γ) at the two tops on the sides which carries it.

See : Squaring n°63 January-March 2007

Tangential triangle with UVW

The U₁ points, U₂ and U₃, intersections of right-hand sides (PQ), (RS) and (MN), are located on the symédianes.

The U₁U₂U₃ triangle is the tangential triangle of UVW, It is homothetic tangential triangle T₁T₂T₃ of ABC in the homothety of center L.

Another construction of the circle starting from M and NR

Starting from a point M of (AB) distinct to lead the parallel to the tangent has some with (Γ). It crosses (AC) in NR. To build the points U₂ and U₃, intersection of (MN) with the symédianes (CL) and (BL). To plot straight lines (RS) and (PQ) parallel with the tangents with (Γ) out of B and C and to find the four other points of the circle.

Definition 3: Construction of three antiparallels equal length

The lines (MN), (PQ) and (RS) are antiparallel at the sides of the triangle and the segments which they determine are of the same length.

This property can be taken as definition by determining three segments, length equal and parallel with the tangents in has, B, C with the circumscribed circle.

Construction

Starting from a point M of (AB) distinct to lead the parallel to the tangent has some with (Γ), therefore perpendicular to (AO). It crosses (AC) in NR. To defer the length MN on the tangent out of B to (Γ) in R₁ and R₂, on the tangent out of C in Q₁ and Q₂. The parallel with (AB) passing by R1 crosses (BC) in R, the parallel with (BC) passing by R2 crosses (AB) in S. the parallel to (AC) passing by Q1 crosses (BC) in Q and the parallel with (BC) passing by Q₂ crosses (AC) in P.

We obtain a configuration of six points, these points cocyclic and are located on a circle of Tücker.

Justification

The parallel with (AB) passing by NR cuts the tangent out of B to (Γ) in R₁ and (BC) a R. By parallelism, the circle circumscribed with triangle MNR recuts the sides of the triangle ABC out of P, Q and S. As one saw in the definition 2, it is a circle of Tücker.

being built, it can be delicate to choose, starting from B, the direction towards R₁ or R₂ to place R.

Medium of the cords, construction starting from a given center

The mediums form a triangle UVW resulting from ABC in a homothety of center L of report/ratio T with |T| = LU/LA. In this homothety, the point O has as an image Ω with LΩ/LO = |T|. This point Ω is the center of the circumscribed circle with UVW. Line (UΩ) parallel with (OA) is perpendicular to (MN), it is the mediator of. In the same way (VΩ) is the mediator of. Ω is well the center of the circle (T).

A circle of Tücker is characterized by its center Ω located on (OL), distinct from O and L.

Construction

The parallel with (OA) passing by Ω crosses (IT) out of U.

M and NR are located on the perpendicular out of U at (OA) and one supplements R by parallelism to build the circumscribed circle with MNR.

Two circles of Tücker

|T| = LU/LA.

By prolonging the sides of the triangle U' V' W' until those of the triangle ABC, we obtain a second circle of Tücker passing by Me P' Q' R'.

By taking the mediums of the cords, the triangle U' V' W' is homothetic ABC in a homothety of center L of report/ratio you: |you| = LU' /LA = |(t+1) /2| (indeed U' is the medium of).

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