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}.
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Date Modified: 2022.0225