- An old problem:

Let ABCD be a quadrilateral. To construct two congruent and tagent circles such that each one be tagent to sides of opposite angles of ABCD.

So we have two cases:

(Ka),(Kc) tagent to sides of A,C, resp.

(Kb),(Kd) tagent to sides of B, D, resp.

My Question:

For which quadrilaterals the four centers Ka, Kb, Kc, Kd are

concyclic? For cyclic ones maybe?

APH - El 18/02/2013 0:26, Antreas escribió:
> An old problem:

A synthetic construction for that circles exist, because its radius can

> Let ABCD be a quadrilateral. To construct two congruent and tagent circles such that each one be tagent to sides of opposite angles of ABCD.

>

> So we have two cases:

>

> (Ka),(Kc) tagent to sides of A,C, resp.

>

> (Kb),(Kd) tagent to sides of B, D, resp.

>

> My Question:

>

> For which quadrilaterals the four centers Ka, Kb, Kc, Kd are

> concyclic? For cyclic ones maybe?

>

> APH

be determined from the vertices coordinates by rational operations and

square roots. But, it's known a simple synthetic one?

--

Saludos,

Ignacio Larrosa Cañestro

A Coruña (España)

ilarrosa@...

http://www.xente.mundo-r.com/ilarrosa/GeoGebra/ - [APH]
> An old problem:

See the construction of the circles here:

> Let ABCD be a quadrilateral. To construct two congruent and

> tagent circles such that each one be tagent to sides of

> opposite angles of ABCD.

>

> So we have two cases:

>

> (Ka),(Kc) tagent to sides of A,C, resp.

>

> (Kb),(Kd) tagent to sides of B, D, resp.

>

> My Question:

>

> For which quadrilaterals the four centers Ka, Kb, Kc, Kd are

> concyclic? For cyclic ones maybe?

http://anthrakitis.blogspot.gr/2013/02/circles-of-quadrilateral.html

Note that we get a quadrilateral center, the point KaKc /\ KbKd,

and a quadrilateral central line, the line joining the points

of contact of (Ka),(Kc) and (Kb),(Kd).

Antreas ----- Original Message -----

From: "Antreas" <anopolis72@...>

To: <Hyacinthos@yahoogroups.com>

Sent: Tuesday, February 26, 2013 10:07 AM

Subject: [EMHL] Re: Quadrilateral Problem

> [APH]

>> An old problem:

>> Let ABCD be a quadrilateral. To construct two congruent and

>> tagent circles such that each one be tagent to sides of

>> opposite angles of ABCD.

>>

>> So we have two cases:

>>

>> (Ka),(Kc) tagent to sides of A,C, resp.

>>

>> (Kb),(Kd) tagent to sides of B, D, resp.

>>

>> My Question:

>>

>> For which quadrilaterals the four centers Ka, Kb, Kc, Kd are

>> concyclic? For cyclic ones maybe?

>

> See the construction of the circles here:

>

> http://anthrakitis.blogspot.gr/2013/02/circles-of-quadrilateral.html

Thanks. I make an applet with GeoGebra:

http://www.xente.mundo-r.com/ilarrosa/GeoGebra/DosCircTg4l.html

based on a construction of Eduardo in es.ciencia.matematicas.

> Note that we get a quadrilateral center, the point KaKc /\ KbKd,

> and a quadrilateral central line, the line joining the points

> of contact of (Ka),(Kc) and (Kb),(Kd).

>

> Antreas

>

>

>