An angle standing on a chord

• 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 4-by-4 count checks pair by pair ×4
• 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
• 1/MX − 1/MY is the same for every pair of chords ×1
• 325 = 5²·13 is a sum of two squares in 24 ordered, signed ways ×1
• 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 hexagon has fourteen triangulations ×1
• a hexagon on a circle has a constant total ×1
• a hexagon well off its circle does not ×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
• across every random butterfly the two distances agree ×1
• AD = cos β ×1
• all six feet are the same distance from the midpoint of P and P* ×1
• among isosceles triangles the margin is smallest at the equilateral one ×1
• and each is as long as the least total distance ×1
• and each product of two still fixes a corner of the trisectors' triangle ×1
• and its incentre is the inner circle's centre ×1
• and its second and third ×1
• and its y coordinate ×1
• and not after three ×1
• and only the flat one is equilateral ×1
• and PA·PB = PC·PD ×1
• and so do the other two ×1
• and so does the third ×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 equilateral triangle is the smallest margin of all ×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 the three pieces make up the side ×1
• and their radii differ ×1
• and touching each escribed one ×1
• as the point nears the circle its conjugate runs away ×1
• at Euler's distance the rotation number is one third ×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
• climbing from random sets of four comes close to the tetrahedron, and for five and six points the climbs stall above the chain ×1
• conjugating twice returns the point ×1
• cos of an odd multiple is invariant and of an even multiple is not ×1
• each junction is 120° ×1
• each pair is two incidences: each point on the other's circle ×1
• each pair of Malfatti circles touches ×1
• each side over the sine of the opposite angle is the diameter ×1
• each trisector cuts its angle in exact thirds ×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 angle of the triangle is below 120° ×1
• every apex on the arc gives the same angle ×1
• every chain of four tangents closes ×1
• every chain of three tangents closes ×1
• every chord through M has the same product of pieces ×1
• every conjugate lies on one conic ×1
• every prime in the best squared distance leaves remainder 1 on division by 4 ×1
• every side of every equilateral one makes (B − C)/3 with the base, give or take 60° ×1
• every side subtends 120° at the point ×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 triangulation of a heptagon has four flips ×1
• every triangulation of the cyclic hexagon has the same total ×1
• every trisector triangle in the sweep is equilateral ×1
• for every triangle the signed distances add to R + r ×1
• for the Fermat point the path is straight ×1
• greedy beats Malfatti on every triangle of the grid ×1
• in both coordinates ×1
• in the corrected coordinate every step advances by the same fraction of a turn ×1
• in the plane both facts hold ×1
• including PQ itself ×1
• inscribed angles on the same chord are equal ×1
• junctions in the hexagon's shortest network ×1
• Newton's method solves Malfatti's three side equations ×1
• no random set goes below √3/2 ×1
• OI² = R(R − 2r) on every triangle ×1
• on the ellipse the wings cross on opposite sides ×1
• on the ellipse the wings cut the chord equally ×1
• on the hyperbolic plane the three sides are not equal ×1
• on the parabola the wings cross on opposite sides ×1
• on the parabola the wings cut the chord equally ×1
• on the sphere the spread grows with the excess while the triangle is small ×1
• on the sphere the three cubed rotations still compose to nothing ×1
• on the sphere the three sides are not equal ×1
• one angle, not several ×1
• one segment per pair ×1
• opposite angles add to a straight angle ×1
• opposite angles of a cyclic quadrilateral add to π ×1
• past five points the ratio falls steadily, towards about 0.784 ×1
• Ptolemy: the diagonals' product is the sum of the products of opposite sides ×1
• pushed off the circle, the totals spread ×1
• random points in a cube are far from the worst case: averages above nine-tenths, minima above the tetrahedron's ×1
• rotating A by 60° about B gives the apex C′ ×1
• so BC / sin A is the diameter ×1
• so do the angles at C and B ×1
• so the circumradius is at least twice the inradius ×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 C stand on the same arc BD ×1
• the angles at A and D stand on the same arc and are equal ×1
• the angles at D and B stand on the same arc AC ×1
• the apex is on the major arc ×1
• the apex lies on the circle ×1
• the broken path C′P′PC has the length of the total distance from P ×1
• the centre's angle is twice the apex's ×1
• the centroid has a larger total than the Fermat point ×1
• the centroid's partner is the symmedian point ×1
• the chain does not close after three steps ×1
• the chain's corners fall in proportion to one over the tangent length ×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 conic passes through O, H and the symmedian point ×1
• the construction and the minimisation find the same point ×1
• the corner C has an angle of at least 120° ×1
• the crossing of the construction lines is no longer the minimum ×1
• the distance satisfies Fuss's relation ×1
• the exterior angle is twice the base angle ×1
• the Fermat views take no options ×1
• the greedy circles cover more of the triangle than Malfatti's ×1
• the grid is between 3 and 14 points on a side ×1
• the hexagon with its centre attains √3/2 ×1
• the hexagon's shortest network has length 3√3 ×1
• the incentre is its own partner ×1
• the incentre is not on that line, which the determinant reports rather than being told ×1
• the incentres make a right angle at every corner ×1
• the inner circle is less than half the outer ×1
• the inner triangle's first two sides are equal ×1
• the iteration finds a network no longer than the spanning tree ×1
• the least total is at the wide corner ×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 lines from corners to opposite apexes are equal ×1
• the marked lengths are realised in many directions ×1
• the meeting point has weights a·sec⁴(A/4) on the corners ×1
• the named quantities split as the angle criterion predicts ×1
• the network with two 120° junctions has length 1 + √3 ×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 orthocentre's partner is the circumcentre ×1
• the pair count by direction agrees with checking every pair ×1
• the pocket beside the second circle never holds the largest third circle ×1
• the point is inside the circle ×1
• the quadrilateral's angles add to 360° ×1
• the regular simplex's ratio falls with every dimension, from √3/2 ×1
• the regular tetrahedron saves 18.7%, and a chain of twelve points along the helix saves more ×1
• the rotation number never increases as the inner circle moves out ×1
• the sample includes obtuse triangles, where one distance counts negative ×1
• the search never goes below √3/2 ×1
• the shape sweep is between 6 and 60 steps on a side ×1
• the short-side pairing wins and the diagonal pairing is worst ×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 spread is rounding error exactly for the invariant functions ×1
• the spread shrinks in proportion to the excess ×1
• the Steiner views take no options ×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 third circle goes deeper exactly when k(α)² beats k(β) ×1
• the third line passes through the same point ×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 three lines meet at one point on every triangle of the sweep ×1
• the three reflected lines are parallel ×1
• the three reflected lines meet where the weights say, a²/x : b²/y : c²/z ×1
• the three unit pulls cancel at the Fermat point ×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 diagonals give the same total ×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 two triangulations agree on Σ cos 3x and disagree on Σ cos 2x ×1
• the two triangulations' inradii add to the same total ×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
• the wings cross on opposite sides of M ×1
• the wings cut the chord at equal distances from its midpoint ×1
• three roads meet at 120° at every junction ×1
• triangle OPA is isosceles ×1
• triangle OPB is isosceles ×1
• triangulations of a heptagon ×1
• two touching circles on one line touch it 2√(r r′) apart ×1
• unequal products put D off it ×1
• up to seven points no branching pattern beats the one that follows the chain ×1
• which passes through the three corners ×1
• while MX − MY itself varies ×1
• with the circles concentric every step turns by the same angle ×1
• Σ cos is constant over the flip graph ×1

The whole library · What the figures prove