Generator

An angle standing on a chord

A generator in the geometry library, called 52 times across 10 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

circle-angle is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

An angle standing on a chord. A circle with a fixed chord and a movable apex on the major arc. The angle at the apex is 60 degrees wherever the apex is put, and the angle the same chord subtends at the centre is 120 degrees.

The classical centres as three weights each

Three centres that always fall on one line

The nine-point circle, touching four others

A quadrilateral inscribed in a circle

Four apexes, one angle

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

at x = 0.00 the signed product is d² − r² ×21
at x = -9.00 the signed product is d² − r² ×20
the distance between points 1 and 2 is the fraction 3/5 ×10
point 1 of the nine is the same distance from the centre ×9
the chord for inscribed angle 10° is 2R sin 10° ×9
the chord at 0° splits into pieces whose product is r² − d² ×4
the half-chord over the split 2 + 8 has square 16 ×4
the apex at 120° sees AB at the same angle ×3
the secant at 180° has near × far = 144 ×3
a corner of a piece is α+60° ×2
a corner of a piece is β+60° ×2
a corner of a piece is γ+60° ×2
each third at A is 26° ×2
each third at B is 18° ×2
each third at C is 16° ×2
the assembled triangle's angle at A is 78° ×2
the assembled triangle's angle at B is 54° ×2
the assembled triangle's angle at C is 48° ×2
a corner of a piece is 60° ×1
a corner of a piece is α ×1
a corner of a piece is β ×1
a corner of a piece is γ ×1
A lies on the circle ×1
a triangle whose incentre is on the Euler line has two equal sides ×1
a trisector and the far side are not parallel ×1
AB = cos α ×1
abc/4K is the radius of the circle through the corners ×1
AD = cos β ×1
and its second and third ×1
and its y coordinate ×1
and PA·PB = PC·PD ×1
and so do the other two ×1
and so is the nine-point centre with two of them ×1
and that distance is half the radius of the circle through the corners ×1
and the angle at C in the diameter's triangle is right ×1
and the centroid sits twice as far from the orthocentre as from the circumcentre ×1
and the incentre stays off the line on every scalene triangle in it ×1
and the same construction on the other trisectors is not ×1
and the same on the other side ×1
and the tangent meets the radius at a right angle ×1
and their radii differ ×1
and touching each escribed one ×1
B and D lie on the circle with diameter AC ×1
B lies on the circle ×1
BC = sin α ×1
BD = sin(α + β): a chord of a unit-diameter circle is the sine of the angle it subtends ×1
CD = sin β ×1
each side over the sine of the opposite angle is the diameter ×1
eighteen of the twenty-seven are equilateral ×1
equal products put D on the circle through A, B and C ×1
equilateral exactly when the three choices do not add to 2 mod 3 ×1
every apex on the arc gives the same angle ×1
every side of every equilateral one makes (B − C)/3 with the base, give or take 60° ×1
every triangle in the sweep has its nine-point circle touching its inscribed one ×1
every triangle in the sweep puts its nine points on one circle ×1
every trisector triangle in the sweep is equilateral ×1
one angle, not several ×1
opposite angles add to a straight angle ×1
Ptolemy: the diagonals' product is the sum of the products of opposite sides ×1
so BC / sin A is the diameter ×1
so do the angles at C and B ×1
the angle at D stands on the same arc BC as the angle at A ×1
the angle between the chord and the tangent equals the inscribed angle ×1
the angle on a diameter is a right angle ×1
the angles add to 180° ×1
the angles at A and D stand on the same arc and are equal ×1
the apex is on the major arc ×1
the apex lies on the circle ×1
the centre's angle is twice the apex's ×1
the chord is a diameter ×1
the circumcentre, centroid and orthocentre are collinear ×1
the circumcentre, the centroid and the orthocentre lie on one line ×1
the collinearity holds across the sweep ×1
the exterior angle is twice the base angle ×1
the incentre is not on that line, which the determinant reports rather than being told ×1
the inner triangle's first two sides are equal ×1
the lines from A and B are not parallel ×1
the lines from B and C are not parallel ×1
the lines from C and A are not parallel ×1
the nine-point centre is the midpoint of that line ×1
the nine-point circle touches the escribed opposite A circle from outside ×1
the nine-point circle touches the escribed opposite B circle from outside ×1
the nine-point circle touches the escribed opposite C circle from outside ×1
the nine-point circle touches the inscribed circle from inside ×1
the point is inside the circle ×1
the quadrilateral's angles add to 360° ×1
the shape sweep is between 6 and 60 steps on a side ×1
the side is 8R sin α sin β sin γ ×1
the side opposite A makes (B − C)/3 with BC ×1
the splitters at A and B are not parallel ×1
the splitters at B and C are not parallel ×1
the splitters at C and A are not parallel ×1
the sweep covers a decent share of the shape space ×1
the sweep found enough non-degenerate triangles to mean something ×1
the sweep found enough triangles ×1
the sweep found enough triangles to mean something ×1
the sweep runs over between 20 and 2000 triangles ×1
the tangent length squared is the same number ×1
the tangent ray on the far side of the chord makes the inscribed angle with it ×1
the three angles are the angles of a triangle ×1
the three angles are those of a triangle ×1
the three angles are those of a triangle and none is a sliver ×1
the triangle has three corners and some area ×1
the triangles are similar, so PA/PD = PC/PB ×1
the two circles are not concentric ×1
the two halves are the whole angle ×1
the two sides agree at every angle ×1
the two sides of a piece are not parallel ×1
the view is one the family draws ×1
the weights for the centroid give its x coordinate ×1
the weights for the circumcentre give its x coordinate ×1
the weights for the incentre give its x coordinate ×1
the weights for the nine-point centre give its x coordinate ×1
the weights for the orthocentre give its x coordinate ×1
triangle OPA is isosceles ×1
triangle OPB is isosceles ×1
unequal products put D off it ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Geometry

A centre is three weights

Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.

Geometry

An angle that does not care where it stands

Fix two points on a circle and look at them from anywhere else on the far arc. The angle is the same from every one of those places, and it is exactly half the angle at the centre.

Geometry

Eighteen equilateral triangles

Every angle of a triangle has three trisectors, not one, once the angle and its outside are both counted. Choosing one at each corner gives twenty-seven ways to cut out a triangle, and eighteen of them give an equilateral one. The nine that fail are exactly the choices whose labels add to 2, 5 or 8 — and all eighteen equilateral triangles have their sides in the same three directions, fixed by a third of the difference between two angles.

Geometry

Every ray comes back to the other focus

An ellipse has two foci and one property everybody remembers: the distances to them add to a constant. What that property forces is stranger and more useful — a mirror shaped like an ellipse sends every ray leaving one focus, in every direction, through the other.

Geometry

Every side measured by one diameter

In any triangle, divide each side by the sine of the angle opposite it: the three answers are equal. That much is the law of sines, and it is usually left there. The common answer has a name — it is the diameter of the circle through the triangle's corners — and the reason is the inscribed angle once more. It is also why the sine was first a half-chord, why Ptolemy's theorem is the addition formula, and why a circle can hold infinitely many points all at rational distances from each other.

Geometry

Nine points on one circle

Three midpoints, three feet of altitudes and three more midpoints. Nine points defined in three unrelated ways, on an arbitrary triangle, and all nine sit on one circle — checked here on two hundred and forty triangles as well as on the drawn one.

Geometry

One circle touching four

The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.

Geometry

One number for every chord through a point

Draw any line through a point and let it cut a circle twice. The two distances from the point to the circle multiply to the same number whichever line is drawn — inside, outside, or grazing as a tangent. The number belongs to the point, and the reason it does not depend on the line is the inscribed angle: two chords through a point cut out two triangles with the same angles.

Geometry

Seven pieces and an equilateral middle

Morley's theorem has two proofs worth knowing, and they run in opposite directions. The trigonometric one starts from the triangle and computes each side of the inner one as 8R sin α sin β sin γ, symmetric in the three angles. Conway's starts from an equilateral triangle, builds six pieces round it from their angles alone, and shows they fit — so the triangle they make is whatever triangle was wanted, and its middle is equilateral because it was built that way.

Geometry

Three trisectors and a triangle nobody expected

Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.

The whole library · What the figures prove