Plates on the geometry of the triangle and the circle, each one computed in the browser as you watch: the circumcentre, the centroid and the orthocentre, which always lie on one line, Euler's, the centroid a third of the way along, and the circle through nine points that every triangle has, which touches its incircle and all three of its excircles; and Morley's theorem, that the trisectors of the angles of any triangle meet in an equilateral triangle, with the seven pieces of Conway's proof and the eighteen equilateral triangles made by trisecting every angle all three ways; and Napoleon's theorem, that equilateral triangles raised outward or inward on the sides of any triangle have centres that form an equilateral triangle, with van Aubel's squares on the sides of a quadrilateral and the three rounds of the same construction that make any pentagon regular; and Simson lines, the feet of the perpendiculars from a point on the circumcircle lying on one line that turns half as fast as the point and wraps Steiner's deltoid, with the plane shaded by the number of Simson lines through each point; and chains of circles, Steiner's porism, in which a ring of touching circles between two circles closes from every start if it closes from one, seen through a map that makes the two circles concentric, and Pappus's chain in the arbelos, the n-th circle n of its diameters above the base. The figure never stops moving, and every point, line and circle is constructed from it again in every frame; the corner measures on the construction the quantity a theorem says cannot change, beside one that does.
图版
I–V
循环
每张 60 秒
分辨率
随播放区域自适应，最高 440 万像素
预先录好的帧
没有
短片
制作中
图版 I · 4 段 · 1:00 循环
The nine-point circle
Draw any triangle and three of its centres fall on one line: the circumcentre O, where the perpendicular bisectors of the sides meet; the centroid G, where the medians meet; and the orthocentre H, where the altitudes meet. Euler proved in 1765 that G lies a third of the way from O to H. Halfway between O and H is the centre N of a circle through nine points of the triangle, with half its circumradius, and in 1822 Karl Wilhelm Feuerbach proved that this circle touches the incircle and each of the three excircles. The triangle keeps moving and every centre is constructed again, as the crossing of two lines; the corner measures the ratios the theorems say cannot change.
图版
说明
The measurement in the corner
The figure on every plate, a triangle or, on the third, a quadrilateral or a pentagon, keeps moving, and everything else is constructed from it again in every frame: a centre is the point where two lines drawn from the triangle cross, a circle passes through points constructed in the same way. The corner measures on that construction the quantity a theorem says is fixed, a ratio, a length or an angle, beside one that keeps changing, and a small plot at the right traces both. However the figure moves, the fixed one does not waver in the sixth decimal place.
One figure is not enough
None of these theorems can be seen in a single drawing, where three points on a line or nine on a circle could always be a coincidence of that one triangle. They are statements about every triangle at once, and a moving triangle shows a little of that: the construction changes shape at every moment, and the coincidence keeps happening.
Constructed, not stored
Nothing on screen is kept from one frame to the next. Each point is found as the crossing of two lines, in double precision, from the corners of the figure at that instant, and each circle from the points it passes through; the third line through a centre and the other six points on the nine-point circle are drawn where they are, not where they ought to be. The deltoid of the fourth plate is not drawn at all: the shader follows the Simson line round the circle for every pixel and counts the times it passes through.