table8


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Let P be a point in the plane of the reference triangle ABC and denote by

cevP : the cevian triangle of P

acvP : the anticevian triangle of P

pedP : the pedal triangle of P

apdP : the antipedal triangle of P

ccvP : the circumcevian triangle of P

cacP : the circumanticevian triangle of P

refP : the reflection triangle of P in the sidelines of ABC

symP : the reflection triangle of P in the vertices of ABC

The following table gives the loci of P such that two of these triangles are parallelogic. Note the frequent occurence of the Kjp cubic K024 and see Table 22 for a generalization.

true means the property is true for any P, L denotes the line at infinity, C denotes the circumcircle.

	ABC	cevP	acvP	pedP	apdP	ccvP	cacP	refP
cevP	sidelines of ABC and antimedial triangle
acvP	sidelines of ABC	sidelines of ABC and a cubic
pedP	L, C	L, C and the quartic Q1	C and a quartic
apdP	L, C	L, C and a quintic	L, sidelines of ABC and polar circle	L, C and K024
ccvP	L, C and K024	C and a septic	sidelines of ABC, C and a bicircular quartic	L, C and K024	L, C
cacP	a sextic	a nonic	sidelines of ABC	L, the quartic Q2 and a nK(X6, X3)	L, C and an octic	C and the quartic Q2
refP	L, C	C and the quartic Q1	C and a quartic	L, C	L, C and K024	L, C and K024	C, the quartic Q2 and K003
symP	not possible	sidelines of the antimedial triangle	sidelines of ABC	L	L, C	L, C and K024	a sextic	L

Notes :

the quartic Q1 is the isogonal transform of a conic.

the quartic Q2 is the isogonal transform of the polar circle.