Sorry, an error occurred while loading the content.

## Re: {Disarmed} [EMHL] Re: A condition for a quadrilateral to be orthodiagonal

Expand Messages
• Dear Cosmin, I like your argument too, although I would like to observe that your Lemma 1 (or at least the part that you need for your final statement) is
Message 1 of 6 , Jul 1, 2009
Dear Cosmin,

I like your argument too, although I would like to observe that your
Lemma 1 (or at least the part that you need for your final statement) is
again more easily phrased in terms of the isoptic cubic. Indeed, the
locus of all points P in the plane of a quadrilateral ABCD such that
its (perpendicular) projections onto the sides of ABCD are co-cyclic is
again the isoptic curve. This actually follows from some simple ``angle
chasing''. Consequently, by the same argument as in in my first e-mail,
the projections of the diagonal point AC \cap BD of a quadrilateral
ABCD onto its sides can be co-cyclic iff ABCD is orthodiagonal.

Eisso

Cosmin Pohoata wrote:
> Dear Eisso,
>
> Your idea is great. As I don't usually work with cubics, it is always
> a joy for me to see their appearance in such rather easy configurations.
>
> Here is my proof of:
>
> > > Problem: Let ABCD be a convex quadrilateral and let P = AC /\ BD, E =
> > > AB /\ CD and F = AD /\BC. Prove that AC \perp BC if and only if
> > > isog_{ABF}(P)=isog_{CDF}(P)=isog_{ADE}(P)=isog_{BCE}(P).
>
> We use the following easy lemma:
> Lemma 1(some 8-pt circle theorem): Let ABCD be a quadrilateral, and
> let P be the common point of its diagonals. Denote by U1, V1, W1, T1
> the projections of P on AB, BC, CD, DA and by U2, V2, W2, T2 the
> intersections of PU1, PV1, PW1, PT1 with CD, DA, AB, BC, respectively.
> Then, ABCD is orthodiagonal if and only if U1, V1, W1, T1, U2, V2, W2,
> T2 are all concyclic.
>
> This can be proved with a quick angle-chase (if I remember correctly):
>
> Now we also have this:
>
> Lemma 2(well-known). If ABC is a triangle and P, Q two (distinct)
> points in plane. P and Q are isogonal if and only if their pedal
> circles coincide.
>
> Combining these two we get our result.
>
> Regards,
> Cosmin
>
========================================
Eisso J. Atzema, Ph.D.
Department of Mathematics & Statistics
University of Maine
Orono, ME 04469
Tel.: (207) 581-3928 (office)
(207) 866-3871 (home)
Fax.: (207) 581-3902
E-mail: atzema@...
========================================
Your message has been successfully submitted and would be delivered to recipients shortly.