What you are looking at
Drag a vertex far enough and the orthocentre leaves the triangle, the circumcentre crosses a side, and the Euler line swings through everything. What does not change is more interesting than what does.
The colours
- Ink (white in dark mode): the triangle, and the classical centres O, G and H.
- Cyan: the nine-point circle, its nine points, and the Euler line.
- Rose: the incircle, its centre I, and the contact triangle.
- Gold: the altitudes and their feet.
Why the nine-point circle has nine points
Switch on the nine-point circle and count the cyan dots. Three are the midpoints of the sides, three are the feet of the altitudes, and three are the midpoints of the segments joining each vertex to the orthocentre. Nine points that have no obvious reason to be concyclic, and yet they are, for every triangle you can drag into existence.
Its centre N is the midpoint of OH, and its radius is exactly half the circumradius. Watch those two facts survive as you deform the triangle.
Euler's theorem, live
The panel shows the distance OI between the circumcentre and the incentre alongside √(R²−2Rr). They agree to three decimal places no matter where you drag the vertices. That is Euler's theorem, and it has a corollary you can also see: since OI² cannot be negative, R is always at least 2r, with equality only for the equilateral triangle. Try to make the two circles concentric and you will find the triangle forced into that one shape.
Where this comes from
These configurations are the subject of my monograph, which treats them as a systematic toolkit rather than a gallery of curiosities.
More olympiad resources About the book Sample chapter: Cross Ratios