- Dear Angel and Chris,

Another construction for this: Let Qi be the cyclocevian conjugate of Pi wrt PjPkPl. QiQjQkQl could be called the 'Cyclocevian Quadrangle' of PiPjPkPl, and its perspector is this new center, which might be called the 'Cyclocevian Center'.

A related center would be the centroid of the Cyclocevian Quadrangle. Coordinates?

Best regards,

Randy Hutson

>________________________________

[Non-text portions of this message have been removed]

> From: Angel <amontes1949@...>

>To: Hyacinthos@yahoogroups.com

>Sent: Friday, June 29, 2012 12:04 PM

>Subject: [EMHL] Re: new centers/items in complete quadrangle and complete quadrilateral

>

>

>

>Dear Chris van Tienhoven,

>

>I have a Quadrangle Center that is not currently in EQF.

>

>Let P1, P2, P3, P4 be the defining Quadrangle Points.

>Let S1 = P1.P4 /\ P2.P3, S2 = P1.P3 /\ P2.P4 and S3 = P1.P2/\ P3.P4.

>Now S1 S2 S3 is the QA-Diagonal Triangle of the Reference Quadrangle.

>

>For each vertex Pi, we take the triangle TjTkTl, where Tj the

>intersection of the sidelinea PkPl with circumcircle of the triangle S1S2S3 (other than S1, S2, S3)

>Qi = Perspector of the triangle PjPkPl and TjTkTl

>

>The "unknown" Quadrangle Center is the common intersection point of lines Pi.Qi

>

>1st CT-Coordinate:

>

>p(q+r)(p+2q+r)^2(p+q+2r)^2

>((1/(p+2q+r)^2)((p+r)(-c^4(p+r)^2(q+r)^2+

>(p+q)^2(a^4(p+r)^2+b^4(q+r)^2))

>((q+r)^2(2p+q+r)^2SA+(p+r)^2(p+2q+r)^2SB+

>(p-q)^2(p+q)^2SC))+

>(1/(p+q+2r)^2)((p+q)(c^4(p+r)^2(q+r)^2+(p+q)^2(a^4(p+r)^2-

>b^4(q+r)^2))((q+r)^2(2p+q+r)^2SA+

>(p-r)^2(p+r)^2SB+(p+q)^2(p+q+2r)^2SC)))

>

>1st DT-Coordinate: a^2(c^4p^2q^2 + (b^4p^2 - a^4q^2)r^2)

>

>Construction GeoGebra:

>

>http://amontes.webs.ull.es/geogebra/EQF_QA_Pn.html

>

>If ABC is the diagonal triangle of the Quadrangle with a vertex in triangle center X(n), Then the Quadriangle Center obtained here is the TCC-perspector of X(n). See the note just before X(1601) in ETC.

>

>Best regards,

>

>Angel Montesdeoca

>

>--- In Hyacinthos@yahoogroups.com, "Chris Van Tienhoven" <van10hoven@...> wrote:

>>

>> Dear Friends,

>>

>> I got some very nice remarks following my announcement of the Encyclopedia of Quadri-Figures.

>> Seiichi Kirikami attended me on a special property of a Complete Quadrangle.

>> The Orthopoles of a line with respect to the four component triangles of any complete quadrangle lie on a straight line known as the Orthopolar Line of for the given complete quadrangle (see also "Orthopolar line" in Mathworld).

>> Definition Orthopole of a random line wrt some Triangle:

>> If perpendiculars are dropped on any line from the vertices of a triangle, then the perpendiculars to the opposite sides from their perpendicular feet are concurrent at a point called the Orthopole (see also "Orthopole" in Mathworld).

>>

>> I tried some things with the Orthopolar Line and found these results:

>> 1. The 4 Orthopoles of a line wrt the Component Triangles in a Complete Quadrangle are collinear. Let's name this the QA-Orthopole Line.

>> 2. The 4 Orthopoles of a line wrt the Component Triangles in a Complete Quadrilateral are also collinear. Let's name this the QL-Orthopole Line.

>> 3. Every QA-Orthopole Line of a line through QA-P4 (Isogonal Center) is a line through QA-P2 (Euler-Poncelet Point).

>> It looks like this is the only case that a pencil of lines through a point is being transformed into another pencil through a point.

>> The locus of their mutual crosspoint is a hyperbola through QA-P2 and QA-P4.

>> Its conic center is the midpoint of QA-P2 and QA-P4.

>> Both lines are parallel when they are parallel to the asymptotes of this hyperbola.

>> 4. A QL-Orthopole Line is always a line perpendicular to the Newton Line (QL-L1). Most Special.

>>

>> I refer here to some points as coded in the Encyclopedia of Quadri-Figures:

>> http://www.chrisvantienhoven.nl/mathematics/encyclopedia.html

>>

>> Are there any (synthetic) proofs for 1., 2., 3. and 4. ?

>> Are there any more related properties known?

>>

>> Best regards,

>>

>> Chris van Tienhoven

>>

>>

>> --- In Hyacinthos@yahoogroups.com, "Chris Van Tienhoven" <van10hoven@> wrote:

>>

>> > You can find this Encyclopedia of Quadri-Figures (EQF) at:

>> > http://www.chrisvantienhoven.nl/mathematics/encyclopedia.html

>> >

>> > The results also can be downloaded in PDF-format at:

>> > http://www.chrisvantienhoven.nl/7-mathematics/191-downloads-eqf.html

>> >

>>

>

>

>

>

>

- I repeat my previous message because of typo in hyperlink.

Dear Friends,

I added a new property wrt a conic in a complete quadrangle in EQF.

See: http://www.chrisvantienhoven.nl/other-quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/198-qa-co-1.html

If any more references are known about this subject please let me know.

Best regards,

Chris van Tienhoven

--- In Hyacinthos@yahoogroups.com, "Chris Van Tienhoven" <van10hoven@...> wrote:

>

> Dear Friends,

>

> I noticed there are all kind of centers, lines, conics, cubics, etc. in a Complete Quadrilateral/Complete Quadrangle.

> So I started writing a paper on this subject.

> Because I discovered so many new items I decided to make a catalogue of them.

> More than 150 items are catalogued now.

> I gave them all a unique code and name.

> * Jean-Louis' Euler-Poncelet Point (Hyacinthos message 19258) is coded with QA-P2.

> * Jean-Pierre's Homothetic Center (Hyacinthos message 19635) is coded with QA-P4.

> * Apart from the well-known quadrangle centroid QA-P1 there is also a quadrilateral centroid QL-P12.

> I placed them all at my site with references where I found them.

> I also found many new items. These you will find without reference.

> When there are more references please let me know.

>

> You can find this Encyclopedia of Quadri-Figures (EQF) at:

> http://www.chrisvantienhoven.nl/mathematics/encyclopedia.html

>

> The results also can be downloaded in PDF-format at:

> http://www.chrisvantienhoven.nl/7-mathematics/191-downloads-eqf.html

>

> Special thanks for Eckart Schmidt who painstakingly checked all items and helped me to eliminate several typos and mistakes and gave me lots of good advice. We had a very nice conversation alternately in German and English.

>

> When you know about other Quadrangle/Quadrilateral items please let me know.

>

> Best regards,

> Chris van Tienhoven

>