Advanced triangle analysis
In-depth analysis of objects with respect to a reference triangle
The advanced triangle analysis is a more sophisticated version of the simple triangle analysis. It is thus an observation technique in which the object under consideration (a point, a line, a circle or a conic) is related to characteristic objects of a reference triangle. It uses a relatively large database of triangle objects.
The advanced triangle analysis on the current construction is performed as a separate module, which is invoked by pressing the (blue) triangle button in the main toolbar or by the command Commands|Triangle Analysis. In the module you specify the object to be analysed, the reference triangle, the desired depth of analysis, the extent of the report, and the type of analysis.
The result of the analysis is a more or less exhaustive report containing a lot of technical terms. You can
- edit and copy the report as a text,
- get definitions and explanations of terms in the report,
- view the objects mentioned in the text or add them to the construction,
- get the related information from the Encyclopedia of triangle centres (ETC).
The types of analysis available in the Advanced triangle analysis are illustrated below. All illustrations examples are performed using the following configuration:
Let N be the centre of the nine-point circle of a given triangle ABC. Let A' be the mirror image of N in the line BC. The points B' and C' are defined cyclically.
In the ifollowing lllustrative analysis, we examine how the triangle A'B'C', its circumcircle and its circumcentre S' are related to the characteristic objects of the triangle ABC. Recall that for each analysis you can specify
- the number of considered ETC centres,
- whether to take into consideration the triangle transformations (e.g., isogonal conjugation),
- the extent of the considered properties and the size of the output,
- how to name centres (e.g. X1 or X1: centre of incircle).
Object analysis
We first consider the point S' with respect to the reference triangle ABC. A limited extend of Object analysis shows that S' is the Kosnita point (X54) of ABC, which is also the isogonal conjugate of the nine-point centre (X5) of ABC - and this is often all one wants to know. On the other hand, an extensive Object analysis of the point S' (Extended size of report with Transformed centres and 16342 centres considered) yields several properties.
Here are some illustrative data from the extensive results:
- S' is the Kosnita point (X54) of ABC; it is also: isogonal conjugate of X5, isotomic conjugate of X311, complement of X2888, etc. (8 items)
- The complement of S' in ABC is the isogonal conjugate of X1166 in ABC; the Ceva conjugate of (S',X3) is the isogonal conjugate of X847 in ABC, etc. (237 items)
- S' lays on the Euler line of the Circumorthic triangle of ABC; S' lays on the Feuerbach line of the Kosnita triangle of ABC, etc. (7 items)
- S' is the insimilcentre of the Sine-triple circle of ABC and the Nine point circle of ABC (1 item)
- S' is the midpoint of: X3 and X195 in ABC, X145 and anticomplement of X7979 in ABC, etc. (139 items)
- S' lays on Jerabek hyperbola of ABC, the iso-conjugate of (S',X5) lays on the Feuerbach hyperbola of ABC, etc. (16 items)
- S' lays on the conic through A, B, C, X2, X24, on conic through A, B, C, X15, X18, etc. (4 items)
- S' lays on lines through X6, X24,..., through X4, X184, etc. (235 item)
- The distances between collinear points: S', X2, X1209 involve the ratio 3/2; the distances between S', X973, X3567 involve the ratio 5/4, etc. (69 items)
- S' is the image of X1 under the projectivity (A,B,C,X5)›(A,B,C,X1), etc. (67 items)
The Object analysis of the circumcircle of the triangle A'B'C' yields less data:
- The circle through A'B'C' contains 3 ETC centres (up to index 16342): X1263 (isogonal conjugate of X1157), X6343 (Hatzipolakis-Lozada-Euler reflections point), X10264 (Hatzipolakis-Moses image of X74).
- The circle through A'B'C' contains 5 transformed ETC centres (up t index 16342): the isogonal conjugate of X1157, the complement of X399, the complement of 13512, the anticomplement of X6592, the anticomplement of X10272.
Comparison of the ETC centres of two triangles
The Two triangles centres analysis, applied to the triangle A'B'C', finds those (transformed) ETC centres of A'B'C', which coincide with any ETC centre of ABC.
Considering the first 16342 centres of ABC and A'B'C', the test quicly finds 4 ETC coinciding centres of the two triangles:
- X2: centroid (A'B'C') = X5946: nine-point center of orthocentroidal triangle (ABC)
- X3: circumcenter (A'B'C') = X54: Kosnita point (ABC)
- X4: orthocenter (A'B'C') = X13368: point Beid 49 (ABC)
- X20: De Longchamps point (A'B'C') = X15532: intersection of lines X(20)X(1154) and X(206)X(576) (ABC)
Perspectivity analysis
The Perspectivity centres analysis (as part of the Object abalysis) of a triangle A'B'C' finds out whether the triangle A'B'C' is perspective to a characteristic triangle of the reference triangle ABC. It considers several types of perspectivity: identity, homothety, perspectivity, orthologycal perspectivity, and cyclological perspectivity. The centres of the found perspectivities are also computed and can be inspected (with Object analysis) as ordinary points named %1, %2, etc.
The perspectivty analysis (Extensive size) finds 81 characteristic triangles of ABC, which are perspective to A'B'C' that yield about 120 different points as their centres.
Here are some examples of perspectivities with respective centres of perspectivity:
- A'B'C' is homothetic to the pedal triangle of X5. The centre of the homothety is the nine-point centere of ABC (X5).
- A'B'C' is perspective to the antimedial triangle of ABC. The centre of perspectivity is the Becrux point 27 of ABC (X11271).
- A'B'C' is orthologically perspective to the tangential triangle of ABC. The respective centres of perspectivity are the X5-Ceva conjugate of X3 (X195 in ABC) and the orthocentre of orthic triangle of ABC (X52).
- A'B'C' is cyclologically perspective to the Kosnita triangle. The respective centres of perspectivity are the focus of the Kiepert parabola of ABC (X110) and the midpoint of the circumcentre X3 of ABC (X3) anf the Parry-Pohoata point X8157 of ABC (the midpoint isnot an ETC centre).
Here is an example of perspectivity. The triangle A'B'C' is perspective to the orthic triangle of ABC. The centre P of perspectivity is the orthocentre of the orthic triangle of ABC (X52).
The triangle A'B'C' is cyclologically perspective to the reference triangle ABC. This means that the circles through A,B,C', through A,B',C and through A',B,C have a common intersection, in our case the ETC centre X1263 of ABC (i.e. the isogonal conjugate of the inverse-in-circumcircle of X54). Also, the circles through A',B',C, through A',B,C' and through A,B',C' pass through a common point, in our case X110 ( the focus of the Kiepert parabola of ABC).
Triangle transformations analysis
This type of observation detects if some points in a configuration are related via some triangle transformation of the reference triangle.
We illustrate this on a generalisation of our original configuration: Let P be an arbitrary point not laying on the sidelines of the reference triangle ABC. Let A' be the miror image of P in the line BC. Define B' and C' cyclically. Finally, let S' be the circumcentre of A'B'C'.
To find how S' is related to P (in terms of triangle transformations in ABC), we perform an Object analysis of the point S' with P as 'additional centre' and taking into consideration the triangle transfromations. The analysis finds that S' is the isogonal conjugate of P in ABC.
To find how S' is related to P (in terms of triangle transformations in ABC), we perform an Object analysis of the point S' with P as 'additional centre' and taking into consideration the triangle transfromations. The analysis finds that S' is the isogonal conjugate of P in ABC.
