In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775:
For a given acute triangle determine the inscribed triangle of minimal perimeter.
The solution is the orthic triangle, with vertices at the base points of the altitudes of the given triangle.
The orthic triangle, with vertices at the base points of the altitudes of the given triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of Fagnano's problem. Fagnano's original proof used calculus methods and an intermediate result given by his father Giulio Carlo de' Toschi di Fagnano. Later however several geometric proofs were discovered as well, amongst others by Hermann Schwarz and Lipót Fejér. These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter.
A solution from physics is found by imagining putting a rubber band that follows Hooke's Law around the three sides of a triangular frame , such that it could slide around smoothly. Then the rubber band would end up in a position that minimizes its elastic energy, and therefore minimize its total length. This position gives the minimal perimeter triangle. The tension inside the rubber band is the same everywhere in the rubber band, so in its resting position, we have, by Lami's theorem,
Therefore, this minimal triangle is the orthic triangle.
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
A right triangle or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle, is a triangle in which one angle is a right angle, i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry.
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In geometry, bisection is the division of something into two equal or congruent parts. Usually it involves a bisecting line, also called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector.
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.
In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and the four circles passing through any three of the four points have the same radius.
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.
In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible or, equivalently, the geometric median of the three vertices. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.
In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions. It is also known as the 1-median, spatial median, Euclidean minisum point, or Torricelli point.
Giovanni Francesco Fagnano dei Toschi was an Italian churchman and mathematician, the son of Giulio Carlo de' Toschi di Fagnano, also a mathematician.
An acute triangle is a triangle with three acute angles. An obtuse triangle is a triangle with one obtuse angle and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.
In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stacked rings of isosceles triangles, arranged within each ring in the same pattern as an antiprism. The resulting shape can be folded from paper, and is named after mathematician Hermann Schwarz and for its resemblance to a cylindrical paper lantern. It is also known as Schwarz's boot, Schwarz's polyhedron, or the Chinese lantern.
