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}.
3-Geometry-Plane-Polygon-Kinds
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Date Modified: 2022.0225