Gossard perspector

In geometry the Gossard perspector (also called the Zeeman–Gossard perspector) is a special point associated with a plane triangle. It is a triangle center and it is designated as X(402) in Clark Kimberling's Encyclopedia of Triangle Centers. The point was named ''Gossard perspector'' by John Conway in 1998 in honour of Harry Clinton Gossard who discovered its existence in 1916. Later it was learned
that the point had appeared in an article by Christopher Zeeman published during 1899 – 1902. From 2003 onwards the Encyclopedia of Triangle Centers has been referring to this point as ''Zeeman–Gossard perspector''.

Definition

Gossard triangle

Let ''ABC'' be any triangle. Let the Euler line of triangle ''ABC'' meet the sidelines ''BC'', ''CA'' and ''AB'' of triangle ''ABC'' at ''D'', ''E'' and ''F'' respectively. Let ''A_gB_gC_g'' be the triangle formed by the Euler lines of the triangles ''AEF'', ''BFD'' and ''CDE'', the vertex ''A_g'' being the intersection of the Euler lines of the triangles ''BFD'' and ''CDE'', and similarly for the other two vertices.
The triangle ''A_gB_gC_g'' is called the Gossard triangle of triangle ''ABC''.

Gossard perspector

Let ''ABC'' be any triangle and let ''A_gB_gC_g'' be its Gossard triangle. Then the lines ''AA_g'', ''BB_g'' and ''CC_g'' are concurrent. The point of concurrence is called the ''Gossard perspector'' of triangle ''ABC''.

Properties

*Let ''A_gB_gC_g'' be the Gossard triangle of triangle ''ABC''. The lines ''B_gC_g'', ''C_gA_g'' and ''A_gB_g'' are respectively parallel to the lines ''BC'', ''CA'' and ''AB''.
*Any triangle and its Gossard triangle are congruent.
*Any triangle and its Gossard triangle have the same Euler line.
*The Gossard triangle of triangle ''ABC'' is the reflection of triangle ''ABC'' in the Gossard perspector.

Trilinear coordinates

The trilinear coordinates of the Gossard perspector of triangle ''ABC'' are
: ( ''f'' ( ''a'', ''b'', ''c'' ) : ''f'' ( ''b'', ''c'', ''a'' ) : ''f'' ( ''c'', ''a'', ''b'' ) )
where
: ''f'' ( ''a'', ''b'', ''c'' ) = ''p'' ( ''a'', ''b'', ''c'' ) ''y'' ( ''a'', ''b'', ''c'' ) / ''a''
where
: ''p'' ( ''a'', ''b'', ''c'' ) = 2''a''⁴ − ''a''²''b''² − ''a''²''c''² − ( ''b''² − ''c''² )²
and
: ''y'' ( ''a'', ''b'', ''c'' ) = ''a''⁸ − ''a''⁶ ( ''b''² + ''c''² ) + ''a''⁴ ( 2''b''² − ''c''² ) ( 2''c''² − ''b''² ) + ( ''b''² − ''c''² )² 3''a''² ( ''b''² + ''c''² ) − ''b''⁴ − ''c''⁴ − 3''b''²''c''²

Generalisations

The construction yielding the Gossard triangle of a triangle ''ABC'' can be generalised to produce triangles ''A'B'C' '' which are congruent to triangle ''ABC'' and whose sidelines are parallel to the sidelines of triangle ''ABC''.

Generalisation 1

This result is due to Christopher Zeeman.

Let ''l'' be any line parallel to the Euler line of triangle ''ABC''. Let ''l'' intersect the sidelines ''BC'', ''CA'', ''AB'' of triangle ''ABC'' at ''X'', ''Y'', ''Z'' respectively. Let ''A'B'C' '' be the triangle formed by the Euler lines of the triangles ''AYZ'', ''BZX'' and ''CXY''. Then triangle ''A'B'C' '' is congruent to triangle ''ABC'' and its sidelines are parallel to the sidelines of triangle ''ABC''.

Generalisation 2

This generalisation is due to Paul Yiu.

Let ''P'' be any point in the plane of the triangle ''ABC'' different from its centroid ''G''.
:Let the line ''PG'' meet the sidelines ''BC'', ''CA'' and ''AB'' at ''X'', ''Y'' and ''Z'' respectively.
:Let the centroids of the triangles ''AYZ'', ''BZX'' and ''CXY'' be ''G_a'', ''G_b'' and ''G_c'' respectively.
:Let ''P_a'' be a point such that ''YP_a'' is parallel to ''CP'' and ''ZP_a'' is parallel to ''BP''.
:Let ''P_b'' be a point such that ''ZP_b'' is parallel to ''AP'' and ''XP_b'' is parallel to ''CP''.
:Let ''P_c'' be a point such that ''XP_c'' is parallel to ''BP'' and ''YP_c'' is parallel to ''AP''.
:Let ''A'B'C' '' be the triangle formed by the lines ''G_aP_a'', ''G_bP_b'' and ''G_cP_c''.
Then the triangle ''A'B'C' '' is congruent to triangle ''ABC'' and its sides are parallel to the sides of triangle ''ABC''.

When ''P'' coincides with the orthocenter ''H'' of triangle ''ABC'' then the line ''PG'' coincides with the Euler line of triangle ''ABC''. The triangle ''A'B'C' '' coincides with the Gossard triangle ''A_gB_gC_g'' of triangle ''ABC''.

Generalisation 3

Let ''ABC'' be a triangle. Let ''H'' and ''O'' be two points, and let the line ''HO'' meets ''BC, CA, AB'' at ''A₀, B₀, C₀'' respectively. Let ''A_H and A_O'' be two points such that ''C₀A_H'' parallel to ''BH'', ''B₀A_H'' parallel to ''CH'' and ''C₀A_O'' parallel to ''BO'', ''B₀A_O'' parallel to ''CO''. Define ''B_H, B_O, C_H, C_O'' cyclically. Then the triangle formed by the lines ''A_HA_O, B_HB_O, C_HC_O'' and triangle ''ABC'' are homothetic and congruent, and the homothetic center lies on the line ''OH''. Dao Thanh Oai, ''A generalization of the Zeeman-Gossard perspector theorem'', International Journal of Computer Discovered Mathematics, Vol.1, (2016), Issue 3, page 76-79
If ''OH'' is any line through the centroid of triangle ''ABC'', this problem is the Yiu's generalization of the Gossard perspector theorem.

References

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Triangle centers