On planes, straight lines intersect at one or no points {plane geometry}|. Two points determine a line. For every straight line, through any point outside the line, one line is parallel to the line. In triangles, angle sum is 180 degrees.
Flat surfaces {plane, geometry} have two dimensions.
Surfaces have extent measures {area, surface} {surface area}.
The point at which two lines intersect has opening {angle, geometry} between lines.
Lines {bisector, angle}| can divide angles in half.
Angles have sides {half-line} {ray, angle}.
Bisectors {internal bisector} can be perpendicular to external bisector.
In tessellations, vertexes {nodal point, tessellation} can be common to three or more polygons.
Circles can divide into 360 equal arcs {degree, angle}. Degrees can divide into 60 parts {minute, angle}. Minutes can divide into 60 parts {second, angle}.
Angle units {grad} {grade, angle} can equal 1/100 right angle.
2*pi radians, or 360 degrees, make one revolution {perigon}.
Circles with one-unit-length radius have circumference 2*pi radians, so 360 degrees equals 2*pi radians {radian, angle}. One radian equals 360 / (2 * pi) degrees or 57 degrees.
Angles {acute angle}| can be less than 90 degrees.
Two angles {complementary angle}| can add to 90 degrees.
Two angles {corresponding angle, plane} can have parallel corresponding sides.
One ray {initial side} can be stationary and one ray {terminal side} can rotate, so angle {directed angle} changes.
Two angles {equal angle} with parallel sides are equal.
Polygon sides make angles {exterior angle} outside polygon vertices. External angle is 360 degrees minus internal angle.
Two polygon sides make an angle {interior angle} {internal angle} inside a polygon vertex.
Angles {obtuse angle}| can be between 90 and 180 degrees.
Angles {quadrantal angle} can be (n * pi) / 2 radians, where n = 0, 1, 2, ..., for 0, 90, 180, ... degrees.
Angles {reflex angle, size} can be greater than pi radians and less than two times pi radians.
Angles {straight angle} can be pi radians.
Two angles {supplementary angle}| can add to 180 degrees.
Two lines intersect to form opposite angles {vertical angle}.
Closed geometric figures {circle, geometry} have centers and circumferences. Circle equation is (x - h)^2 + (y - k)^2 = r^2, where r is radius. For given perimeters, out of all plane figures, circles bound greatest area {area, circle}. Circle area = pi * r^2, where r is radius.
Plane figures {annulus}| can have ring shape.
Circles can have parts {arc, circle} greater than semicircles {major arc} or less than semicircles {minor arc}. Arcs subtend central angle. Area subtended by circle arc = 0.5 * r^2 * A, where A is angle in radians, and r is radius. Arc length s equals radius r times angle A in radians: s = r*A. In two different circles, for same angle, arc-length ratio is proportional to radii ratio.
Circles can have angles {central angle} between two radii.
Line segments {chord, circle}| can link two circle points. Diameter is longest chord.
2*pi radians equals 360 degrees {circular measure}. pi radians equals 180 degrees. pi/2 radians equals 90 degrees.
distance around circle {circumference}.
Circles have radius-length reciprocal {curvature, circle}.
All curve points can invert {inverse curve}. Operations {inversion, curve} can find inverse curves.
For polygons, circumscribed-circle radius {long radius} is longer than inscribed-circle radius.
Circle perimeter is 2 * pi * radius {perimeter, circle}.
Circles can have inscribed quadrilaterals, which have two diagonals. Diagonal product equals sum of opposite-side products {Ptolemy's theorem} {Ptolemy theorem}.
Circle regions {quadrant, circle}| can include one-quarter circumference and two radii.
Two intersecting lines intersect a circle to make line segments {rectangular properties}.
Chords {secant, circle}| can extend beyond circles.
Circular arc and radii from endpoints make pie-piece figures {sector, circle}| {segment, circle}. Sectors {major sector, circle} {major segment, circle} can be greater than semicircles. Sectors {minor sector, circle} {minor segment, circle} can be less than semicircles.
Diameter ends define half circle {semicircle}|. Angle from circle point to diameter ends is right angle.
Arcs define {subtend}| central angle. Area subtended by arc is 0.5 * r^2 * A, where r is radius and A is arc length in radians. Arc length s equals radius r times angle A in radians: s = r*A. In two different circles, for same angle, arc-length ratio is proportional to radii ratio.
Solids {zone, sphere} {band}| can result when circle sector rotates around sphere diameter that does not pass through sector. Zone is on sphere surface. Right-circular cone connects sphere center to closer zone base.
Solids {cap, circle}| {cap zone} can result when circle sector rotates around sphere diameter that passes through sector. Right-circular cone connects sphere center to cap base.
Circles can make ellipsoids {spheroid} of revolution.
Two imaginary points {circular point at infinity} on line at infinity are common to two circles.
Several points {concyclic point} can lie on same circle.
Line from point intersects circle at two points, to form secant. Distances from point to both points can multiply {power, point}. Point power is negative if point is inside circle and positive if point is outside circle.
For circles, radius midpoints define smaller-circle {inversion circle} centers that intersect first circle at only one point and include larger-circle center. Radius diameters intersect first circles at points {inverse point}. Distance from first intersection to first-circle center times distance from inverse point to first-circle center equals r^2 {inversion constant}.
Circles {concentric circle} can have same center.
Circles {escribed circle} can touch three consecutive polygon sides, if two polygon sides extend.
Circles {great circle}| on spheres can have same radius as sphere.
Equation (x - a)^2 + (y - b)^2 + c^2 = 0 has radius = i*c {imaginary circle}.
Circle can touch three consecutive polygon sides {incircle} {inscribed circle}. Inscribed circle has center {incenter}.
Two circles {orthogonal circle} can intersect at right angles. Curves {orthogonal trajectory} can intersect all curve-family members at right angles.
Plane intersects sphere to make circle {small circle}. Small circle does not include great circle.
Line segments can be straight or not-straight {curve}.
Curves have parts {arc, curve}.
At large positive or negative values, curves can approach straight lines {asymptote, curve}.
Equations with parameters can define sets of curves {family of curves}.
Linkages {Peaucellier's linkage} {Peaucellier linkage} can construct inverse curves.
Figures have length around edges {perimeter, curve}.
Curved line-segment length {rectification of curve}| is integral from x = x1 to x = x2 of (1 + (dy/dx)^2)^0.5 * dx.
Moving curves or surfaces can have contact points or lines with fixed curves or surfaces, with no slippage {rolling motion}. Moving curve rotates around centroid. Centroid revolves around fixed-surface curvature center at contact point or line.
Straight lines can intersect curves to form line segments {secant, curve}|.
Curve parts {segment, curve} lie between two points. Area between arc and chord makes segment. Solid formed by two parallel planes intersecting sphere makes segment.
Normals at curve points can make orthogonal projections {subnormal, curve} onto x-axis.
Tangents at curve points can make orthogonal projections {subtangent} onto x-axis.
Chords {supplemental chord} can go from any circle or ellipse point to diameter end.
Function points {acnode} {isolated point} can have no nearby points that belong to function. Acnodes must be at least double points.
Points {centroid, curve} can be curve-point-coordinate arithmetic means.
Curve branches meet at a point {multiple point}.
Two curve branches can meet at point {node, curve} {double point}. Both branches can have real and distinct tangents {crunode}. Both branches can have real and coincident tangents {cusp, curve}. Both branches can have imaginary tangents at acnodes.
Curves can have double cusps {tacnode} {point of osculation} {osculation point}.
Turning points or maximum or minimum points {stationary point} do not have to be inflection points.
Wires can have a shape {brachistochrone} so that a bead can slide from one end to the other in shortest possible time.
Equation y = a * e^(b*x) defines curve {exponential curve}|.
Points or envelopes can slide along two fixed curves and make curve {glissette}.
Curve families {integral curve} can be differential-equation solutions.
For successive c values, function f(x,y) = c makes curves {isoclinal}. If dy/dx = f(x,y), solutions have graphs.
Lines {Jordan curve} can have no multiple points. Plane closed curves have inside and outside {Jordan curve theorem}.
Two curves {parallel curves} can have same normal, have same curvature center, and be in one-to-one correspondence.
Continuous curves {Peano-Hilbert curve} can fill continuous two-dimensional regions. Start with square. Put four squares inside and connect centers to make polygonal curve. Put four squares inside small squares to make 16 squares and connect centers to make polygonal curve. Continue to make polygonal curve that approaches passing through all points inside starting square.
Hippias of Elias [-430 to -420] invented a curve {quadratrix curve} {trisectrix}. Begin with semicircle and line segment equal to radius but tangent to semicircle at end. Move line segment through semicircle uniformly, keeping direction, until it is tangent to other semicircle end. Simultaneously rotate ray, starting from same semicircle end as line segment, around semicircle until it goes through other semicircle end. Line segment and ray intersect to form a curve.
Curves {quadric curve} can have second-order equations. Surfaces {quadric surface} can have second-order equations.
Continuous curves {Riemann-Weierstrass curve} can have no direction at all points. Start with straight line segment with slope +1 and straight line segment with slope -1, meeting in middle, like this: ^. Then, on line segments, repeat construction, to make shape like M, with four line segments, placing base endpoints half as far apart as before. If repeated, both endpoints approach same point, and slopes approach infinity.
Curves {sine curve} can look like waves.
Curves {sinusoid} {sinusoidal curve} can be like sine curves.
Curves {tangent curve} can have same tangent at points.
Graphs {tangent graph} can look like third-power functions.
x^3 + x * y^2 + a * y^2 - 3 * a * x^2 = 0 defines curves {trisectrix of Maclaurin}, which can trisect angles.
One continuous xy-plane curve {unicursal curve, continuous} can make x and y be finite continuous functions of a parameter.
Curves {envelope, curve}| can be tangential to all curve-family curves. Curve families can have equation f(x,y,a) = 0, for parameter a. Curve families have boundary curves. Eliminating parameter from function and taking partial differential with respect to parameter can find envelope.
Curve normals {evolute} can form envelopes. Curve evolute is curvature-center locus.
Epicycloids {cardioid} can be curves traced by one circle point rolling on fixed equal-radius-circle outside: r = 2 * a * (l - cos(A)), where r is distance from pole, a is fixed-circle radius, pole is where rolling point meets fixed circle, and A is angle to radius. Cardioids have one loop and are special limaçon-curve cases.
Circle points rolling along straight lines make curves {cycloid}. Circle points rolling on fixed-circle outsides make curves {epicycloid}. Circle points rolling on fixed-closed-curve outsides make curves {pericycloid}. Circle points rolling on fixed-circle insides make curves {hypocycloid}. Fixed circles can have four times rolling-circle diameter {astroid}.
Conchoids {limaçon of Pascal} can have circle for fixed curve: r = 2 * a * cos(A) + b, where r is distance from pole, pole is where rolling point meets fixed circle, A is angle to radius, and a and b are constants. If b < 2*a, limaçon of Pascal has two loops. If b > 2*a, limaçon of Pascal has one loop. If b = 2*a, limaçon of Pascal is a cardioid curve.
Curve points or line envelopes rolling along fixed curves make curves {roulette curve}.
Oval points can have distances to two fixed points. Product of distances can be constant {Cassini oval}. Product equals b^2, where b is distance when oval point is equidistant from fixed points.
Secant endpoints can make curves {cissoid}.
Two points on the line passing through fixed point and intersecting fixed curve both maintain equal and constant distance from intersection point and make two curves {conchoid}. Fixed curve is asymptotic to both conchoid branches.
Conchoids {conchoid of Niomedes} can have straight lines as fixed curves. Conchoid of Niomedes can trisect angle and duplicate cube.
Points on flexible but not stretchable thread, kept taut while being wound or unwound on another curve, can trace a curve {involute}. Curve used for winding is traced-curve involute.
For Cassini ovals, distance between both points can be constant {lemniscate curve}.
Feet of perpendiculars from rectangular-hyperbola center to all tangents makes a curve {lemniscate of Bernoulli}. If fixed point is circle radius times square root of two from circle center, and two points are secant-through-fixed-point chord length from fixed point, both point loci make lemniscate of Bernoulli.
Point maintaining same distance to intersection point for all tangents to fixed curve makes curve {orthotomic curve}.
Points where perpendiculars from fixed points meet tangent to fixed curve point {pole, pedal curve} make curves {pedal curve}. For parabolas, pedal curves are tangents at vertices. For ellipses or hyperbolas, pedal curves are auxiliary circles. In pedal curves, perpendicular length relates to length from fixed point to curve point {tangent-polar equation} {pedal equation}.
Point sets {spiral} around a fixed point {center, spiral} can have distance {radius vector, spiral} from center to fixed point that relates to rotation angle {vectorial angle}.
equiangular
Spirals {equiangular spiral} can have equal inclinations of radius vector and tangent vector at all points.
types
Archimedes spiral is r = a*A, where a is constant, A is angle, and r is radius. Spirals {parabolic spiral} {Fermat's spiral} can be r^2 = a*A.
Equiangular spirals {logarithmic spiral} {logistic spiral} can be log(r) = a*A.
reciprocal
Spirals {hyperbolic spiral} can be: r*A = a. Hyperbolic spirals are same as inverse {reciprocal spiral}.
loxodromic spiral
Spirals {loxodromic spiral} {rhumb line, spiral} in spheres can cut meridians at constant angle.
Straight lines can pass through fixed points {pole, strophoid} to intersect all fixed curve points. Two line points maintain same distance to intersection as distance from intersection to another fixed point, not necessarily on line, to make a curve {strophoid}.
Curve points can have distance to a fixed point. Fixed point has distance from coordinate origin. Curves {tractrix} can have constant ratio of distance from curve point to fixed point to distance from origin to fixed point.
For radius-a circle with center at (0,a), a straight line from the origin intersects the circle to define the y-coordinate for all x {witch of Agnesi} {Agnesi witch} {versed sine curve} {versiera} (Maria Agnesi): y = 8 * a^3 / (x^2 + 4 * a^2).
Curves {logistic curve} can model growth with growth factors and limiting resources. Growth can depend on growth factors, which can have weights. Limiting resources {bottleneck, growth} can slow growth.
Amount or percentage P at time t is P(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)), where A = constant growth-factor weight, m = original growth-factor amount, n = original limiting-resource amount, and T = time period or inverse growth rate.
Growth is percentage of factor to resource. Denominator and numerator decrease at same rate.
beginning
Original amount depends on relative m and n amounts and on weight A: P(0) = A * (1 + m) / (1 + n). If n is much greater than m and A, denominator is large, and P is zero.
process
At first, growth is exponential with time. Then growth passes through time when growth has constant rate and is linear. Then growth slows exponentially to zero. Amount is then maximum or 100 percent at maturity.
shape
Logistic curve has sigmoid shape.
comparison
Logistic function inverts natural logit function.
For numbers between 0 and 1, functions {logit curve} can be logit(p) = log(p / (1 - p)) = log(p) - log(1 - p). Logarithmic base must be greater than 1, for example, 2 or e. p / (1 - p) is the odds, so logit is logarithm of odds. Logistic function inverts natural logit function. Logit functions can be linear {logit model}: logit(p) = m*x + b. Logit models {logistic regression} can be for linear regression.
Inverse cumulative distribution functions, or normal-distribution quantile functions, depend on error-function inverses {probit curve}. It changes probability into function over real numbers: probit(p) = 2^0.5 * erf^-(2*p - 1). Probit functions can be linear over a large real-number middle range {probit model}.
Logistic curve can look like S {sigmoid curve}| {standard logistic function}. Sigmoid curve starts at minimum or maximum, always increases or decreases, and ends at maximum or minimum. It has one inflection point, near which it grows linearly. It changes exponentially at beginning and end. Amount or percentage P over time t is P(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)). If A = 1, m = 0, n = 1, and T = 1, P(t) = 1 / (1 + e^-t). dP/dt = P * (1 - P), with P(0) = 1/2 and dP(0) / dt = 1/4.
Arctangent, hyperbolic tangent, and error function make sigmoid curves.
Curves {double sigmoid curve} can look like double S. Double sigmoid curves start at minimum or maximum, always increase or decrease, and end at maximum or minimum. They have two inflection points, near which it grows linearly. Growth rate is zero in middle. Double sigmoid curves change exponentially at beginning and end and near middle. Amount or percentage N over variable x is N(x) = (x - d) * (1 - exp(-((x - d) / s)^2)), where d is average, and s is standard deviation.
Logistic curves {Verhulst curve} can have growth rate directly related to current percentage or total amount and directly related to current resource amount. dN / dt = r * N * (1 - N/K), where N is total population, r is growth rate, and K is maximum population possible {carrying capacity, logistic}. Amount or percentage N over time t is N(t) = (K * N(0) * e^(r*t)) / (K + N(0) * (e^(r*t) - 1)), which comes from N(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)), where A ~ 1, m ~ N(0), n ~ N(0) / K, and T = -1/r.
Lines {line} are straight and have no endpoint.
Line segments can connect {join} two points.
Dividing line segment internally or externally can divide second line segment in same proportions {proportional division}. Put second-segment end on first-segment end. Then draw straight line between other endpoints. Then draw line parallel to straight line at first-line-segment dividing point.
Figures {rectilinear figure, line} can have only straight lines.
Line-segment multiples can have greater length than any line segment {Archimedes axiom, line} {axiom of Archimedes, line}.
Plane-geometry axioms {Euclid's twelfth axiom} {Euclid twelfth axiom} can claim that parallel lines exist.
One and only one straight line through a point not on another straight line does not intersect second straight line {parallel postulate}|.
Euclid's twelfth axiom can be simpler {Playfair's axiom} {Playfair axiom}. Through any point, only one line can be parallel to fixed line. Given straight line and exterior point, only one straight line is parallel to the line.
Three lines can meet at angles of 120 degrees {isometric axes}.
Lines can intersect coordinate planes at one point {piercing point}, where they orthogonally project onto coordinate planes. Surfaces intersect coordinate planes to make curves {trace curve}.
Four symmetry axes {quartic symmetry} can make eight angles of 45 degrees.
A line {transversal} can cross two coplanar lines, to form four angle pairs. Angles can orient in same direction {corresponding angle, transversal}. Angles {alternate angle} can lie on opposite sides of coplanar-line transversal. Angles can be opposite each other at intersection points {vertically opposite angle}.
If two transverse lines cross parallel lines, transversal line-segment ratios are equal {intercept theorem}.
Pappus invented line and centroid theorems {Pappus's theorems} {Pappus theorems}.
Draw line segments from first first-line-segment point to first second-line-segment point, second first-line-segment point to second second-line-segment point, and first first-line-segment point to second second-line-segment point. The drawn line segments have three intersection points that make a straight line.
Arcs have planes. Arcs rotated around plane axis generate area equal to arc length times circular-path segment traveled by arc centroid.
Planes rotated around plane axis generate volume equal to surface area times circular-path segment traveled by centroid.
revolution
If centroid paths are circles, the previous two theorems generate surfaces or volumes of revolution.
Two straight lines {anti-parallel line} can cut two straight lines so first angle cut by one line is supplementary to second angle cut by other line.
Line segments {extended line} can be longer, at either end.
Curved lines {isotopic line} can be perpendicular to themselves.
A line {median, trapezoid} joins midpoints of non-parallel trapezium or trapezoid sides.
Figure lines {nodal line} can stay fixed during rotation or deformation.
Lines {oblique line} can be neither parallel nor perpendicular to a direction.
Many-sided figures {polygon} can surround one connected space. Polygons {convex polygon} can have all interior angles less than 180 degrees. Polygons {concave polygon} can have at least one angle more than 180 degrees. Polygons {circumscribed circle} can have all vertices on a circle. Polygons {circumscribed polygon} can have all sides tangent to a closed curve.
Line segments {apothem} {shorter radius} can run from regular-polygon center perpendicular to side.
Three lines, joining opposite vertices of a conic circumscribed hexagon, pass through one point {Brianchon's theorem} {Brianchon theorem}.
Regular polygons can have parallel equal-length sides {opposite side, polygon}. Regular polygons can have opposite equal angles {opposite angle}. Regular polygons can have opposite vertices {opposite vertex}.
A number {order, polygon} of polygons can share a point {nodal point, polygon}.
Figures can have three parts {ternary}.
Figures can have four parts {quaternary}.
Figures can have five parts {quinary}.
Figures can have six parts {senary}.
Figures can have seven parts {septenary}.
Figures can have eight parts {octenary}.
Figures can have eleven parts {undenary}.
Removing a smaller parallelogram, which shares parts of two adjacent sides, from a larger parallelogram makes a figure {gnomon}|.
The five regular-pentagon diagonals make a five-point star {pentagram of Pythagoras}|.
Rectilinear plane figures {polyomino} can involve congruent squares that share sides. Finite numbers of identical squares can join at edges to make shapes, such as crosses or lines. One polyomino can tile plane periodically or not. One polyomino cannot tile plane aperiodically. Polyomino pairs or triples can tile plane periodically, aperiodically, or not. Because aperiodic tilings are possible, no algorithm can decide, for all sets, if a polygon set will tile plane.
Polygons {regular polygon} can have all angles equal and all sides equal.
Games {tangram} can use squares cut into seven pieces, which rearrange without overlapping to make designs.
Polygons {trigon} can have three sides.
Polygons {tetragon} can have four sides.
Figures {pentagon}| can have five sides.
Figures {hexagon}| can have six sides.
Figures {heptagon}| can have seven sides.
Figures {octagon}| can have eight sides.
Figures {dodecagon}| can have 12 sides.
Figures {icosagon}| can have 20 sides.
Polygons {n-gon} can have n sides. In polygons, exterior-angle sum equals 360 degrees. In polygons, interior-angle sum is (n - 2) * (180 degrees).
Figures {quadrilateral}| can have four sides. Quadrilaterals {cyclic quadrilateral} can have all four vertices on a circle.
area: parallelogram
Parallelogram area = b * a * sin(A), where a and b are side lengths and A is angle between them.
area: rectangle
Rectangle area = l*h, where l and h are side lengths.
area: rhombus
Rhombus area = s * s * sin(A), where s is side length and A is small angle.
area: square
Square area = s^2, where s is side length.
area: trapezoid
Trapezoid area = a * sin(A) * (b1 + b2) / 2, where a is vertical-side length, A is small angle between side and base, and b1 and b2 are bases. Trapezoid area = h * (b1 + b2) / 2, where h is height and b1 and b2 are bases.
perimeter: rectangle
Rectangle perimeter = 2*a + 2*b, where a and b are side lengths.
perimeter: rhombus
Rhombus perimeter = 4*s, where s is side length.
perimeter: square
Square perimeter = 4*s, where s is side length.
perimeter: trapezoid
Trapezoid perimeter = a + b + c + d, where a, b, c, d are side lengths.
perimeter: parallelogram
Parallelogram perimeter = 2*a + 2*b, where a and b are side lengths.
Figures {quadrangle}| can have four vertices. Quadrangles {simple quadrangle} can have four vertices and four lines, with no diagonals. Quadrangles {complete quadrangle} can have four points, four lines, and two diagonals.
Diamond-shaped quadrilaterals {kite} can have two equal-side pairs and two equal-angle pairs. Diagonals are perpendicular.
Figures {parallelogram}| can have two pairs of equal, opposite, and parallel sides.
Figures {rectangle} can have two pairs of equal and opposite sides at right angles.
Figures {rhombus}| {rhom} can have four equal sides.
Figures {square figure} can have four equal sides at right angles.
Figures {trapezoid}| {trapezium} can have only one pair of parallel opposite sides.
Quadrilaterals {skew quadrilateral} can have four points not in same plane.
Plane figures {triangle} can have three sides.
area
Triangle area equals 0.5 * b * h, where b is base and h is height.
Triangle area = r*s, where r is inscribed-circle radius, s is (a + b + c) / 2, and a, b, c are sides.
Triangle area = c^2 * sin(A) * sin(B) / (2 * sin(C)), where c is side length, and A, B, C are opposite angles to sides a, b, c.
Triangle area = 0.5 * b * c * sin(A), where b is base length, c is side length, and A is angle between base and side.
area: isosceles
Isosceles-triangle area = 0.5 * b * a * sin(A), where b is base length, a is equal-side length, and A is base angle.
area: equilateral
Equilateral-triangle area = 3^(0.5) * s / 2, where s is side length.
angle sum
Triangle angle sum is 180 degrees.
triangle perimeter
Triangle perimeter = a + b + c, where a, b, c are side lengths. Isosceles-triangle perimeter = 2*a + b, where a is equal-side length, and b is other-side length. Equilateral triangle perimeter = 3*s, where s is side length.
Triangles {congruent}| can be the same but have different locations. Congruent triangles have same three sides, same two angles with same side, and same two sides with same angle.
Three integers {Heronic triple} can represent triangle sides for triangles with integer area.
Triangle area = (s * (s - a) * (s - b) * (s - c))^0.5, where s = 0.5 * (a + b + c) and a, b, c are sides {Hero's formula} {Hero formula}.
For triangles, a circle {nine-point circle} can pass through side midpoints, feet of perpendiculars to sides, and midpoints of line segments between orthocenter and triangle vertices. Nine-point circle center is equidistant to orthocenter and circumcenter.
Right triangles have one right angle. In Euclidean geometry, for right triangles, sum of squares of two shorter sides equals hypotenuse squared {Pythagorean theorem}: c^2 = a^2 + b^2.
proof
To prove theorem, use geometric construction. Use only straightedge and compass to draw new lines and angles. See Figure 1.
Square sides. See Figure 2.
Add original triangle of size 0.5 * a * b, triangle of size 0.5 * a * b beside it, and rectangle of size a*b to squares of sides, to make square of sum of sides and complete the square: (a + b)^2. See Figure 3. (a + b)^2 = a^2 + b^2 + a*b + 0.5 * a * b + 0.5 * a * b = a^2 + b^2 + 2*a*b.
Flip hypotenuse square into square of sum of sides. See Figure 4. c^2 + 4 * (0.5 * a * b) = (a + b)^2. c^2 + 2*a*b = a^ + 2*a*b + b^2. c^2 = a^ + b^2. Hypotenuse squared equals sum of squares of two shorter sides.
For three points, distance between first two points is less than or equal to sum of distance between first and third point and distance between second and third point {triangle inequality}|.
To find side length, first measure base line, then measure angles to other point, and then compute side length {triangulation, length}|. To find angle, first measure base line, then measure sides, and then compute angle {chain triangulation}.
To find space position, first measure distance to three reference points, then find intersection of three spheres {trilateration}|. Global Positioning System (GPS) uses 24 fixed satellites and trilateration by timing signals.
Triangles have line segment {altitude}| from vertex perpendicular to opposite side.
Right triangles have two shorter sides {arm, triangle} {leg, triangle}.
Triangles have a side {base, triangle} intersected by the altitude.
Right triangles have a longest side {hypotenuse, triangle}|.
Triangles have line segments {median, triangle} from vertices to opposite-side midpoints.
Inside triangles, lines from vertexes can meet at two points {Brocard point} and form equal angles at intersections with sides.
Triangle circumscribed circles have centers {circumcenter} inside triangle.
Three medians intersect at one point {median point}.
Triangles have a point {orthocenter} where three altitudes intersect.
Triangles {acute triangle} can have largest angle less than 90 degrees.
Triangles {equiangular triangle} can have all angles equal 60 degrees.
Triangles {equilateral triangle} can have all sides equal.
Triangles {isosceles triangle}| can have two sides equal.
Triangles {obtuse triangle} can have largest angle more than 90 degrees.
From fixed point, lines to vertexes can be perpendiculars {pedal triangle}. Pedal triangles are lines {pedal line} {Simpson's line} if fixed point is on circumscribed circle.
Right triangles {Pythagorean triangle} can have integer-length sides, such as 3, 4, and 5 {rope stretcher's triangle, Pythagorean triangle}; 5, 12, and 13; or 8, 15, and 17.
Triangles {right triangle}| can have one angle of 90 degrees.
Right triangles {rope stretcher's triangle} can have side lengths 3, 4, and 5.
Triangles {scalene triangle} can have no two sides equal.
Triangles {similar triangle} can have same ratios of sides. Similar triangles have corresponding sides and angles.
Triangles {spherical triangle} on spheres can have three right angles {trirectangular spherical triangle} or two right angles {birectangular spherical triangle}.
Different-shape polygons can cover planes {tiling, geometry} with no gaps and no overlaps.
shapes
One triangle, square, or hexagon shape can tile. One pentagon shape cannot tile.
Pairs of shapes can tile, such as two different irregular pentagons. Such tilings have repeated parallelograms {periodic tiling}.
Spirals can tile without repeated parallelogram {aperiodic tiling}. For example, nine-sided triangle-shaped tile {versatile} can tile periodically and non-periodically. Square-shaped tiles with protruding points and corresponding concave depressions can tile aperiodically. Other four-sided shapes {Penrose tiles} have protrusions and depressions, have five-fold symmetry, make repeated patterns, and can tile aperiodically {quasi-periodic tiling}.
periodicity
Algorithms can decide if tiles can tile the plane periodically. No algorithm can decide generally if tiles can tile the plane aperiodically.
Identical shapes, such as triangles, rectangles, hexagons, or special five-sided polygons, or shape sets can fill planes, polyhedrons, or curved surfaces without gaps or overlaps {tessellation}|.
types
Tessellation {regular tessellation} can use equilateral triangles, squares, or regular hexagons. Tessellation {homogeneous tessellation} {semiregular tessellation} can have congruent common vertices {nodal point, homogeneous tessellation} and regular polygons. Tessellation {non-homogeneous tessellation} can use irregular shapes, different sizes of one shape, or both.
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Date Modified: 2022.0225