Triangle Center Loci

Introduction — Thales Circle Triangle Centers on the Thales Configuration

This explorer follows a moving triangle in the Thales configuration: \(A=(1,0)\) and \(B=(-1,0)\) stay fixed while \(C=(\cos\theta,\sin\theta)\) slides around the unit circle \(x^2+y^2=1\). Each choice of \(\theta\) creates a new triangle, and the chosen ETC center \(X_n\) traces a locus.

JSXGraph live viewer Thales configuration ETC centers X(1)–X(99) Euclidean constructions

What you are seeing

Think of this page as a synchronized geometry lab: the orange point moves, the triangle updates, and a classical center such as the incenter, centroid, or symmedian point responds instantly.

  • The blue segment \(AB\) is fixed and acts like a stable ruler.
  • The orange point \(C\) is the "input" of the experiment.
  • The gold trace is the path swept out by the selected center \(X_n\).

Why Thales makes life cleaner

Because \(AB\) is a diameter of the circumcircle, Thales' theorem gives a right angle at \(C\). So every non-degenerate triangle has \(\angle C=90^\circ\).

\[a=|BC|=2\left|\cos\frac{\theta}{2}\right|,\quad b=|CA|=2\left|\sin\frac{\theta}{2}\right|,\quad c=2.\]

Those side lengths are exactly the data used in the barycentric formulas that define many triangle centers.

From classical geometry to coordinates

The README's main idea is that moving centers can be rewritten as rational coordinate formulas, so loci can be computed efficiently and plotted smoothly.

\[t=\tan\frac{\theta}{2},\qquad(\cos\theta,\sin\theta)=\left(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2}\right).\]

That substitution turns trigonometric motion into rational motion, useful for symbolic computation.

Deep dive — a Brilliant-style tour of triangle centers

One of the nicest surprises in Euclidean geometry is that very different definitions can produce points that move in a coordinated way. Some centers are defined by distance, some by balancing, and some by angle symmetry, but in this widget they all respond to the same moving vertex \(C\).

\[P=(u:v:w)\quad\longmapsto\quad\left(\frac{u-v+w\cos\theta}{u+v+w},\frac{w\sin\theta}{u+v+w}\right)\]

Here \((u:v:w)\) are barycentric weights relative to triangle \(ABC\). The converter translates a classical center into a visible Cartesian point.

  • Incenter \(X_1\): balances the side lengths — barycentrics \(a:b:c\).
  • Centroid \(X_2\): average of the vertices — barycentrics \(1:1:1\).
  • Nine-point center \(X_5\): tracks the midpoint geometry of the triangle.
  • Symmedian point \(X_6\): squared side lengths — barycentrics \(a^2:b^2:c^2\).
Euler line moment

The circumcenter is \(O=(0,0)\), the orthocenter is \(H=C\), the centroid is \(G=C/3\), and the nine-point center is \(N=C/2\). Several famous centers line up automatically.

Synchronous motion

When \(C\) rotates once, some centers rotate the same direction, some in the opposite direction, and some collapse to especially simple curves such as circles.

Secondary-school bridge

You can read the picture in two ways at once: as compass-and-straightedge geometry and as analytic geometry on the coordinate plane.

How to read the widget

1. Pick a center. Use the dropdown to jump among ETC entries. The line below the selector shows the active barycentric recipe and whether the locus is tagged as a circle, point, or more general curve.

2. Move \(\theta\). Slide the angle or press Start. Watch the orange vertex move and compare the triangle with the center trace.

3. Switch viewpoints. The coordinate toggle reveals either barycentric area weights or trilinear distance-to-side interpretations.

4. Use the construction panel. For selected centers, the step list gives a Euclidean-construction storyline that matches the moving picture above it.

C—
Xn—
a = |BC|—
b = |CA|—
∠A—
∠B—
\(A,B\) fixed \(C\) on the slider \(X_n\) and its locus circumcircle Euler line \((O\!-\!H)\)
Coordinates
Construction
Barycentric coordinates
—
Select a center to see its coordinates.

⚗ Euclidean Construction

Step 0 / 0

Select a center to see its Euclidean construction.