## Homothetic triangles

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• Let ABC, A B C be two homothetic triangles and X,Y,Z three collinear points. Are the triangles bounded by the lines (AX, BY, CZ) and (A X, B Y, C Z)
Message 1 of 8 , Apr 30, 2010
Let ABC, A'B'C' be two homothetic triangles
and X,Y,Z three collinear points.

Are the triangles bounded by the lines
(AX, BY, CZ) and (A'X, B'Y, C'Z) perspective?

APH
• ... As Francisco pointed out: this follows inmediately from Desargues theorem for any ABC and A B C We can use it in the reference triangle ABC to get
Message 2 of 8 , May 1, 2010
--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
>
> Let ABC, A'B'C' be two homothetic triangles
> and X,Y,Z three collinear points.
>
> Are the triangles bounded by the lines
> (AX, BY, CZ) and (A'X, B'Y, C'Z) perspective?

As Francisco pointed out: "this follows inmediately from Desargues
theorem for any ABC and A'B'C'"

We can use it in the reference triangle ABC to get points,
from two given triangles and three collinear points.

Here is an example:

Inside ABC exists a unique point P such that:
Three Congruent (equal) circles (A'),(B'),(C') concur at P,
and circle (A') touches the sides of the angle A,
(B') of angle B and (C') of angle C.
[A', B', C' are the centers of the circles]

Now, let Lp a line passing through P and
intersecting the circles (A'),(B'),(C') at
X,Y,Z resp. (other than P).

The triangles bounded by the lines
(AX, BY, CZ) and (A'X, B'Y, C'Z) are perspective.

Which is the locus of the perspectors
as L moves around P?

APH
• ... I get two such points P. If your 3 circles have radius r*R/(R+r) then P=X(55). And if their radius is r*R/(R-r) then P=X(56). -- Barry Wolk
Message 3 of 8 , May 6, 2010
Antreas wrote:
> Here is an example:
>
> Inside ABC exists a unique point P such that:
> Three Congruent (equal) circles (A'),(B'),(C') concur at P,
> and circle (A') touches the sides of the angle A,
> (B') of angle B and (C') of angle C.
> [A', B', C' are the centers of the circles]

I get two such points P. If your 3 circles have radius r*R/(R+r)
then P=X(55). And if their radius is r*R/(R-r) then P=X(56).
--
Barry Wolk
• ... inside triangle (that s what I had in mind) APH [Non-text portions of this message have been removed]
Message 4 of 8 , May 6, 2010
On Thu, May 6, 2010 at 11:18 PM, Barry Wolk <wolkbarry@...> wrote:

>
>
> Antreas wrote:
> > Here is an example:
> >
> > Inside ABC exists a unique point P such that:
> > Three Congruent (equal) circles (A'),(B'),(C') concur at P,
> > and circle (A') touches the sides of the angle A,
> > (B') of angle B and (C') of angle C.
> > [A', B', C' are the centers of the circles]
>
> I get two such points P. If your 3 circles have radius r*R/(R+r)
> then P=X(55). And if their radius is r*R/(R-r) then P=X(56).
> --
> Barry Wolk
>
>
> Indeed. Two points P. But I think only one point with circles centers
inside triangle (that's what I had in mind)

APH

[Non-text portions of this message have been removed]
• Let ABC be a triangle and A B C the cevian triangle of I. Denote: Bc, Cb = the reflections of B, C in CC , BB , resp. Oa = the circumcenter of ABcCb.
Message 5 of 8 , Jul 25, 2015
Let ABC be a triangle and A'B'C' the cevian triangle of I.

Denote:

Bc, Cb = the reflections of B, C in CC', BB', resp.

Oa = the circumcenter of ABcCb. Similarly Ob, Oc.

OaObOc and the excentral triangle IaIbIc are homothetic.

Which point is the homothetic center wrt triangles:

1. ABC (on its OI line)

2. OaObOc (on its Euler line)

3. IaIbIc (on its Euler line)  ??

APH
• [APH]: Let ABC be a triangle and A B C the cevian triangle of I. ... Locus: Let ABC be a triangle P a point and PaPbPc the antipedal triangle of P. Denote:
Message 6 of 8 , Jul 25, 2015

[APH]:

Let ABC be a triangle and A'B'C' the cevian triangle of I.

Denote:

Bc, Cb = the reflections of B, C in CC', BB', resp.

Oa = the circumcenter of ABcCb. Similarly Ob, Oc.

OaObOc and the excentral triangle IaIbIc are homothetic.

Which point is the homothetic center wrt triangles:

1. ABC (on its OI line)

2. OaObOc (on its Euler line)

3. IaIbIc (on its Euler line)  ??

APH

Locus:

Let ABC be a triangle P a point and PaPbPc the antipedal triangle of P.

Denote:

Bc, Cb = the reflections of B, C in CP, BP, resp.

Oa = the circumcenter of ABcCb. Similarly Ob,Oc.

Which is the locus of P such that OaObOc, PaPbPc are perspective?

APH

• [APH]: Let ABC be a triangle and A B C the cevian triangle of I. Denote: Bc, Cb = the reflections of B, C in CC , BB , resp. Oa = the circumcenter of ABcCb.
Message 7 of 8 , Jul 26, 2015

[APH]:

Let ABC be a triangle and A'B'C' the cevian triangle of I.

Denote:

Bc, Cb = the reflections of B, C in CC', BB', resp.

Oa = the circumcenter of ABcCb. Similarly Ob, Oc.

OaObOc and the excentral triangle IaIbIc are homothetic.

Which point is the homothetic center wrt triangles:

1. ABC (on its OI line)

2. OaObOc (on its Euler line)

3. IaIbIc (on its Euler line)  ??

APH

1. ABC (on its OI line)
X(3576)

2. OaObOc (on its Euler line)
X(381)

3. IaIbIc (on its Euler line)
X(381)

CL

• Lemma: Let x, y be two perpendicular lines and L a line. The reflections of L in x, y are parallels. Corollary: Let ABC be a triangle, x,y,z three lines
Message 8 of 8 , Nov 1, 2017
Lemma:

Let x, y be two perpendicular lines and L a line.
The reflections of L in x, y are parallels.

Corollary:

Let ABC be a triangle, x,y,z three lines perpendicular to BC, CA, AB, resp. and L1, L2, L3 three lines.

Denote:

La, Lx = the reflections of L1 in BC, x, resp.
Lb, Ly = the reflections of L2 in CA, y, resp.
Lc, Lz = the reflections of L3 in AB, z, resp.

AaBbCc = the triangle bounded by La, Lb, Lc, resp.
AxByCz = the triangle bounded by Lx, Ly, Lz, resp.

AaBbCc, AxByCz are homothetic.

Application:

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

L1, L2, L3 = the Euler lines of AB'C', BC'A', CA'B', resp.

La, Lb, Lc = the reflections of L1, L2, L3 in BC, CA, AB, resp.
Li, Lii, Liii = the reflections of L1, L2, L3 in PA', PB', PC', resp.

AaBbCc = the triangle bounded by La,Lb,Lc
AiBiiCiii = the triangle bounded by Li, Lii, Liii

Which is the locus of the homothetic center of AaBbCc, AiBiiCiii as P moves on a line, the Euler line for example?

APH
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