Sorry, an error occurred while loading the content.

## Quadrilateral Problem

Expand Messages
• 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
Message 1 of 4 , Feb 17, 2013
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
• ... A synthetic construction for that circles exist, because its radius can be determined from the vertices coordinates by rational operations and square
Message 2 of 4 , Feb 19, 2013
El 18/02/2013 0:26, Antreas escribió:
> 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

A synthetic construction for that circles exist, because its radius can
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] ... See the construction of the circles here: http://anthrakitis.blogspot.gr/2013/02/circles-of-quadrilateral.html Note that we get a quadrilateral
Message 3 of 4 , Feb 26, 2013
[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

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
• ... From: Antreas To: Sent: Tuesday, February 26, 2013 10:07 AM Subject: [EMHL] Re: Quadrilateral Problem
Message 4 of 4 , Feb 26, 2013
----- 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
>
>
>
Your message has been successfully submitted and would be delivered to recipients shortly.