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Andrius: I am studying how triangle centers manifest a language by which meaning arises. I am analyzing examples from the Encyclopedia of Triangle Centers. I want to model that with active inference.

One idea for modeling the unconsciously answering mind (by which issues come to matter) and the consciously questioning mind (by which meaning arises) is to study how they interact in studying a particular triangle (with specified lengths of sides and sizes of angles) which the answering mind operates on, while the questioning minds refers to general concepts applying to all triangles. Error arises because while the answering mind executes perfectly, yet its observations have a margin for error. Now for particular triangles there may be results (namely, coinciding triangle centers) which are not true of triangles in general. This gives rise to meaningful statements about particular triangles. These statements can be suggested by the answering mind. But they need to be verified and established by the questioning mind. This can be done interactively or this process can be guided by the third investigatory mind of consciousness. My thought is to model this with Active Inference.

Triangle Centers

Vocabulary

• The Encyclopedia of Triangle Centers has a glossary related tables, and a collection of links
• See also: Mathworld: Triangle and Triangle, List of triangle topics, Encyclopedia of Triangle Centers.
• Clark Kimberling. Major Centers of Triangles.
• Philip J. Davis. The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-History.

Purpose of This Language

The purpose of this language is to make sense of any random triangle, and thus, to interact with a triangle.

Our interaction reflects our three minds.

• We can take actions with regard to the triangle.
• We can construct concepts as a result of those actions.
• We can make and verify statements about those concepts, and in particular, arrive at the same concept with different sequences of actions.

The last point illustrates William James's definition, "Intelligence is the ability to reach the same goal by different means."

I expect the basic actions to accord with six basic cases. These basic actions together interactively define what it means to be a triangle.

The language allows for partial knowledge. Given an action, such as encompassing a triangle with a circle, we can have full knowledge of the triangle by knowing where all three vertices intersect. But we can also have partial knowledge.

• We can not know where the vertices are. But we still know something about the size.
• We can know where a single vertex is.
• We can know where two vertices are. This restricts the third vertex to the longer arc unless they are both equal.
• We can know where all three vertices are.

We can consider the triangle in terms of paths, lines, angles and oriented areas.

Consider the purpose of each action

• Draw the circumcircle. Determine the absolute size of the triangle.
• Draw the incircle. Determine the proportion of the sides.
• Draw the meidan. Construct a fractal coordinate system.
• Draw the orthocenter. Construct the altitudes?

List of Actions

Given a triangle, three points (generally not all on the same line) connected by three lines.

Choose

• Choose a point, line, angle, circle or shape.

Construct a circle

• Given a shape: Draw a circle around the shape. Continuously make it smaller.

Construct a circle and a triangle

• Given a shape: Draw a circle within the shape. Continuously make it bigger. This yields three touch points.

Construct a line

• Given an angle: Divide the angle in half.
• Given two points: Draw a line through the two points.
• Given a point and a line segment: Draw a perpendicular line segment from the point to the line. (When possible.)

Construct a point

• Given a line segment: Divide the line segment in half.
• Given three lines intersecting at a point, an angle between two of the line: Reflect the third line across the angle bisector.
• Given connected line segments: Combine the line segments, fold them in half, yielding the semiperimeter point.

Balance a triangle

Analysis of Triangle Centers

A triangle center is defined as the concurrent intersection of three lines which are defined in terms of the intrinsic properties of the triangle (notably the relative lengths of sides, but independent of the triangle's placement or scale or context). This can be thought of as the three minds coming together to focus on a single point. Triangle centers form a multiplicative group for which the incenter is the identity.

ID and Name	Definition	Set of concepts	Interpretation	Notes
X(1) incenter	A) Equidistant from the three sides. B) Internal angle bisectors intersect.	A) Geodesics (of equal length), points of intersection, arcs of circle. B) Bisected angles.	A) Place a circle inside. Expand it to its maximum size. B) Fold the angles to bisect them. Intersect the resulting lines.	Why do these yield the same center?
X(2) centroid	A) Lines from points to medians. B) Arithmetic mean position of all of the points on the triangle	A) Medians (bisected sides) B) Weight of sides	A) Fold each side to create medians. Draw lines from points to medians. B) Discover the center of mass	Why do these yield the same center?
X(3) circumcenter	A) Center of the circle including the three vertices. B) Intersector of the perpendicular bisectors of the sides	A) Circle and arcs B) Lines and right angles	A) Place circle around the triangle. Shrink the circle to its minimum size. B) Bisect the sides. Draw perpendicular lines at the bisectors.	Why do these yield the same center?
X(4) orthocenter	intersection of the three altitudes		A) draw lines perpendicular to each line, slide each to a vertex equivalently: draw a cross, align a side, draw a line down from the vertex
X(5) nine-point center	center of the nine-point circle, which includes the midpoint of each side, the foot of each altitude, and the midpoint from each vertex to the orthocenter		A) given any three of the nine points, all distinct, find the circumcenter X(3) B) Draw 3 excircles. Draw the circle tangent to all three. Find its center.
X(6) symmedian point	intersection of the three symmedians (which reflect a median across the angle bisector)	median, angle bisector, reflection, intersection with side	fold lines to get medians, fold angles to get bisectors, fold lines over bisectors to get symmedians from medians	Why do these three lines intersect at a point?
X(7) Gergonne point	A) intersection of lines from vertices to touch points of the incircle B) symmedian point of triangle of touch points		A) Given X(1), draw lines from the vertices to the touch points B) If the vertices are not available, given the touch points, build X(6)
X(8) Nagel point	intersection of lines from each vertex to the corresponding semiperimeter point		Combine line segments find semiperimeter point
X(9) Mittenpunkt	symmedian point of the triangle formed by the centers of the three excircles		Construct 3 excircles. Note their centers. Construct triangle. Get X(6).
X(10) Spieker center	center of the Spieker circle	cleaver	A) Draw midpoints of sides. Draw triangle. Get X(1). B) Draw 3 midpoints. Draw 3 semiperimeter points. Connect with 3 cleavers. The 3 intersect. C) Give mass to the perimeter. The center of mass.
X(11) Feuerbach point	Where the incircle and nine-point-circle are internally tangent to each other.	internal	Draw the incircle C(1). Draw the nine-point-circle C(5). Find the point they share inside the triangle.
X(12)	Let A'B'C' be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AA', BB', CC' concur in X(12).		Draw X(6). Draw excircles. Find 3 touch points. Link touch points with vertices. They concur.
X(13) 1st Isogonic Center Fermat point	Point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible.		Around each side of the triangle, draw an equilateral triangle using the side. A) Draw lines from each vertex of an equilateral triangle to the opposite vertex in the original triangle. They concur. B) Draw C(3) around each triangle. Link each X(3) with the opposite vertex. They concur.
X(14) 2nd Isogonic Center		equilateral triangle	Use each side (BC) for an equilateral triangle (A'BC) facing into the triangle. Draw lines A'A, B'B, C'C. They concur.
X(15) first isodynamic point		Appolonius circle equal distance facing outward, inward	At each vertex, note the ratio of distances to the other two vertices, and draw the (Appolonius) circle that maintains this ratio. The three circles meet at two points or one point. Use each side (BC) for an equilateral triangle A'BC facing inward. Draw a reflected triangle ABC facing outward. The lines A'A, B'B, C'C concur.
X(16) second isodynamic point			Same procedure as for X(15). The point outside the triangle. Draw the equilateral triangles facing outward and the reflected triangles facing inward.
X(17) first Napoleon point		centroid of equilateral triangle (center of its incircle)	For each side BC construct an equilateral triangle A'BC facing outward. Let the centroids of this triangles be A. The lines AA, BB, CC concur.
X(18) second Napoleon point			For each side BC construct an equilateral triangle A'BC facing inward. Let the centroids of this triangles be A. The lines AA, BB, CC concur.
X(19) Clawson point
X(20) de Longchamps point
X(21) Schiffler point
X(22) Exeter point
X(39) Brocard midpoint
X(40) Bevan point
X(175) Isoperimetric point
X(176) Equal detour point

Idea

• If there is a sequence of estimatinns by which the center of a triangle is found, then there can be a replay to understand what the estimations should have been. And there might also be an interesting sequence of good estimations. And that sequence is found by learning. And that adds meaning to this activity.

https://math.stackexchange.com/questions/4254480/question-about-the-encyclopedia-of-triangle-centers

https://github.com/lejean2000/Kimberling/