[alogo] 1. Definition of S-triangles

The discussion here is a continuation of the one started in Orthopole.html and Orthopole2.html . We examine triangles which depend on a point P lying inside the deltoid-envelope (see Deltoid.html ) of the Wallace (alias Simson) lines of the triangle of reference ABC.

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[1] Through every point P inside the deltoid pass three tangents to it. These are the Wallace lines {WR,WS,WT} of three points {R,S,T} on the circumcircle of triangle of reference ABC.
[2] The sides {RS,ST,TR} are respectively orthogonal to the Wallace lines {WT,WR,WS}, equivalently the altitudes of RST are parallel to these lines.
These facts are proved in Orthopole2.html . Triangle RST is called an S-triangle with respect to ABC. Here I adopt a naming convention from Lalesco (see ref. [4]) who first investigated these triangles and found most of their properties.

[alogo] 2. Basic angle relation

Fix an orientation on the circumcircle. The sum of oriented arcs AT + BR + CS = 0.
Inversely, for every system of points {T,R,S} on the circumcircle satisfying this relation, the corresponding triangle TRS is an S-triangle i.e. the Wallace lines {WT,WR,WS} of its vertices with respect to ABC are orthogonal to the opposite sides {RS,ST,TR}.

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The proof of the first assertion follows trivially from the way the Wallce line, WT say, is defined (see SimsonProperty.html ): Taking the other intersection point A' of the parallel to WT and drawing A'T orthogonal to BC. Thus, angle(AA'T) has its sides orthogonal to the lines (BC,RS). Draw a parallel BB' to RS and see the relation of arcs AT = -CB'.
To prove the inverse note that the condition implies the orthogonality of RS and WT. Repeating the argument with vertices A and B we obtain the orthogonality of the other pairs of lines too.
This property has a very important corollary since the relation on the three arcs is symmetric with respect to the two triangles ABC and RST.

Corollary If RST is an S-triangle with respect to ABC then later is also an S-triangle with respect to RST (we say that the two triangles are S-related).

Remark The arc condition reveals the symmetry in the following relation which is proved in Orthopole2.html :
If a Wallace line WT is orthogonal to a chord RS, then also the Wallace lines {WR,WS} are orthogonal to the respective chords {ST,TR}.

[alogo] 3. Wallace lines relation

The Wallace lines {W,W'} of a point Q with respect to ABC and an S-triangle RST are parallel.
This is characteristic of the S-relation between triangles, in the sense:
Triangle RST is an S-triangle of ABC if and only if the Wallace lines {W,W'} of the two triangles for every point Q of their circumcircle are parallel.

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This follows from the arc relation of the previous section. For this draw two parallel lines {AA',TT'} and the orthogonals to {BC,RS} respectively from {A',T'}. Use the arc relation to show that these lines intersect at a point Q of the circumcircle. This implies that the Wallace lines {W,W'} of the two triangles with respect to Q are parallel. The argument can be reversed, even in a stronger form to prove:

Proposition Two triangles ABC and RST inscribed in the same circle are S-related to each other if and only if, there is a point Q of this circle with respect to which the corresponding Wallace lines are parallel.

Remark The proposition needs one point Q having the parallelity property of the respective Wallace lines but this implies that the property holds for all points of the common circumcircle.

[alogo] 4. The orthocenters

Using the notation of the previous sections the following properties are valid:
[1] The Wallace line of T w.r to RST is the altitude parallel to the Wallace line WT of T w.r. to ABC.
[2] The orthocenters {H,H'} of ABC and RST are symmetric with respect to P.
[3] The Wallace lines of points {A,B,C} with respect to triangle RST pass also through P, the point defining RST.

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First property follows from the fact that the altitude of every triangle is the Wallace line of the vertex on that altitude. Thus, this altitude and WT are both orthogonal to line RS. It follows that {WR,WS,WT} are parallel to the altitudes of RST. By the symmetry of the S-relation the altitudes of ABC are parallel to the Wallace lines of {A,B,C} with respect to RST. This proves [3].
The symmetry of the orthocenters is a consequence of the other two properties and the fact that HT is bisected by the corresponding Wallace line WT (see SteinerLine.html ).

[alogo] 5. The equivalence relation

S-relation is an equivalence relation among all triangles inscribed in that circle. Among all triangles of an equivalence class there is a distinguished, the mean or derivative equilateral triangle A'B'C'.

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This equilateral triangle was constructed and discussed in Deltoid.html . It results by taking first {A'',B'',C''} to be the intersections of the medial lines respectively to {BC,CA,AB} with the circle lying on the same side with the opposite vertices {A,B,C}. Then taking on the oriented arcs {A''A,B''B,C''C} points {A',B',C'} dividing these arcs in ratio 1:2. An easy angle chasing argument shows that these points define indeed an equilateral triangle and that the arc relation holds true: AA' + BB' + CC' = 0.
Since two different equilaterals inscribed in the same circle cannot satisfy the above condition we conclude that:

Proposition Two triangles {ABC, RST} inscribed in the same circle are S-related if and only if they have the same mean or derivative equilateral.

[alogo] 6. Translated deltoids

The deltoids enveloping the Wallace lines of two S-related triangles ABC and RST are translations of each-other. The translation vector equals the vector between the centers of their corresponding Euler circles. The orthopoles {X,Y} of a tangent tQ with respect to the two triangles map to each other by means of this translation.

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This property follows immediately from the way these deltoids are generated as hypocycloids discussed in Deltoid.html . The orthopoles {X,Y} are the contact points of the parallel Wallace lines of Q with respect to the two triangles. Hence they are corresponding points under this translation.

There is a particular position of Q on the circumcircle for which the two corresponding parallel Wallace lines with respect to ABC and RST coincide. This and some additional properties of S-Triangles are examined in Orthopolar.html .

[alogo] 7. References

[1] Gallatly William The modern geometry of the triangle London, Francis Hodgson 1913, p.49.
[2] Goormaghtigh, R. On Some Loci Connected with the Orthopole-Geometry The American Mathematical Monthly, Vol. 37, No. 7. (Aug. - Sep., 1930), pp. 370-371.
[3] Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington DC, Math. Assoc. Ammer., 1995, pp. 106-110.
[4] Lalesco, T. La Geometrie du Triangle. Paris, Jacques Gabay, 1987, p. 17.
[5] Ramler, J. O. The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle The American Mathematical Monthly, Vol. 37, No. 3, (Mar., 1930), pp. 130-136
[6] Ramler, J. O. On Triangles Having a Common Mean The American Mathematical Monthly, Vol. 47, No. 3, (Mar., 1940), pp. 140-145

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