ADGEOM 5028 * ADGEOM 5029

#5028

Dear friends, "Circumconics with asymptotes making a given angle" is the title of my recent article published in Forum Geometricorum

 

An update can be found at my personal blog. This update introduces a new center, the focus of a parabola:

 

Briefly, the locus of perspectors of circumconics whose asymptotes make a given angle is a conic. This conic becomes a parabola for some particular angle related to Brocard angle. The focus F of this parabola is the point I introduce. Some properties of this point are:

 

First barycentric coordinate: a^2*(2*a^6 - b^6 - c^6 - 3*a^4*(b^2 + c^2) - 6*a^2*(b^4 - 3*b^2*c^2 + c^4)) : :

 

ETC search numbers: {1.10017518584542021, 4.49336719756577036, 0.0220986439717287574}

 

On lines X(i)X(j) for these {i,j}: {{6,110},{126,3589},{141,6719},{182,14688},{187,1084},{511,14650},{518,11721},{524,5914},{543,597},{1296,5085},{1428,3325},{1503,5512},{1576,21309},{2330,6019},{2492,2780},{3618,14360},{5027,6088},{5050,11258},{5166,9019},{5480,23699},{6094,11166},{6096,21448},{10748,14561},{14654,14853},{14666,20423}}

 

Combos: {X[126]-2*X[3589], X[141]-2*X[6719], 2*X[182]-X[14688], X[1296]-3*X[5085], 5*X[3618]-X[14360], 3*X[5050]+X[11258], X[10748]-3*X[14561], X[14654]+3*X[14853], X[14666]+X[20423]}

 

Midpoint of X(i) and X(j) for these {i,j}: {{6,111},{14666,20423}}

 

Reflection of X(i) in X(j) for these {i,j}: {{126,3589},{141,6719},{14688,182}}

 

Best regards,

Francisco Javier Garcia Capitan

 

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#5029

Hi Francisco Javier,

 

Another property of this point: it is the radical trace of circles {{X(6),X(13),X(16)}} and {{X(6),X(14),X(15)}}.

 

Best regards,

Randy Hurson