geometry

A triangle from its three vertices, with the corners lettered. The letters are what this is for: polygon draws the outline in one line already, and then every figure wants three dots and three letters placed beside them without landing on the sides. Each letter is a point's label, so the graph's own label layout keeps it clear of the ink.

A right-angled triangle standing on its corner, the square marked and each side labelled — the figure trigonometry is taught on. The legs are LENGTHS rather than opposite corners: a runs right from the corner and b runs up from it, so the right angle is where the two meet and cannot be put anywhere else. The sides are drawn as three segments rather than as one polygon because each one carries a label, and a label belongs to the side it names.

The Pythagoras figure: a right triangle with a square drawn outward on each of its three sides, each square labelled with its area. The hypotenuse's square is the one nobody wants to work out by hand. Its far corners are the near ones displaced by the hypotenuse turned a quarter turn — $(-a, b)$ becomes $(b, a)$ — which is why the figure comes out as four coordinates and not as a construction.

A regular polygon of any number of sides, about a centre. Drawn side by side with @for, because a series is text the compiler reads in one piece and a loop generates elements rather than entries: the last side closes the shape because its far corner is the turn come all the way round. turn rotates the whole figure — a square at 45° stands on a corner, and at 0° it sits flat. It also derives its interior angle, so a lesson can state the number beside the figure without writing it twice: name the call and read #id.interior. The angles read sides and nothing else, which is what lets them be worked out as the document compiles — sides drives the @for, so it is a compile-time position that can never hold a value waiting on the reader, whereas radius and turn are parametised and so are not arithmetic the compiler can finish. The lengths go through the first side the figure draws instead, which follows a slider on radius.

The point halfway between two others, lettered, with the segment it halves drawn behind it. The segment is drawn as its two HALVES so that ticks can mark them equal: one tick on each is the notation for "these two lengths are the same", and it is the whole reason a midpoint is worth marking rather than stating.

A segment, its midpoint, and the perpendicular bisector through it — the locus of the points equidistant from the two ends. The bisector runs length either side of the midpoint along the segment's normal, which is the segment turned a quarter turn and divided by its own length. Written out that division is the whole construct: it is what keeps the arms the length they were asked for however long the segment is.

An arc of a circle, from one bearing to another, both in degrees anticlockwise from the right. A parametric curve rather than a run of points: the arc is exact at every zoom, and there is no vertex count to pick. fill-style shades what the arc encloses on its own, which is the SEGMENT of the circle — the piece the chord between its ends cuts off. The slice from the centre is sector, because that region closes somewhere else. circle-arc rather than arc because a define may not take a library node's name, and arc is one — the great-circle overlay on the geo canvas.

A sector: the arc between two bearings and the two radii that close it, with the angle at the centre marked. fill-style shades the slice. The arc is drawn here rather than handed to circle-arc for exactly that reason: a sector's region closes through the CENTRE, and an arc left to itself closes across its own chord — which is a different shape and a different piece of the circle.

A circle with its centre dotted and one radius drawn out to a bearing, labelled. size on the circle mark IS the radius, so the segment ends exactly on the circumference — which is the one thing a hand-written version of this figure tends to get wrong.

A tangent touching a circle at a bearing, with the radius to the point of contact and the right angle between them — the circle theorem, drawn.

The unit circle with one angle taken round it: the radius to the point, the cosine along the axis and the sine standing on it, each in its own colour. The two legs are what the figure is for — $\cos\theta$ is the distance across and $\sin\theta$ the distance up, and drawn in different colours they stop being two entries in a table.

Two parallel lines cut by a transversal, with the corresponding angles marked at both crossings — the figure every angle rule is read off. The lower line is $y = 0$ and the upper is $y =$ gap; the transversal crosses the lower one at the origin and the upper one gap / tan along, which is where the second angle is marked. Both angles are measured from the same direction and labelled the same, because that is the claim the picture makes.