In plane geometry, an automedian triangle is a triangle in which the lengths of the three medians (the line segments connecting each vertex to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different order. The three medians of an automedian triangle may be translated to form the sides of a second triangle that is similar to the first one.
The side lengths of an automedian triangle satisfy the formula 2a2 = b2 + c2 or a permutation thereof, analogous to the Pythagorean theorem characterizing right triangles as the triangles satisfying the formula a2 = b2 + c2. That is, in order for the three numbers a, b, and c to be the sides of an automedian triangle, the sequence of three squared side lengths b2, a2, and c2 should form an arithmetic progression. [1]
If x, y, and z are the three sides of a right triangle, sorted in increasing order by size, and if 2x < z, then z, x + y, and y − x are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5, 12, and 13 can be used to form in this way an automedian triangle with side lengths 13, 17, and 7. [2]
The condition that 2x < z is necessary: if it were not met, then the three numbers a = z, b = x + y, and c = x − y would still satisfy the equation 2a2 = b2+ c2 characterizing automedian triangles, but they would not satisfy the triangle inequality and could not be used to form the sides of a triangle.
Consequently, using Euler's formula that generates primitive Pythagorean triangles it is possible to generate primitive integer automedian triangles (i.e., with the sides sharing no common factor) as
with and coprime, odd, and to satisfy the triangle inequality (if the quantity inside the absolute value signs is negative) or (if that quantity is positive). Then this triangle's medians are found by using the above expressions for its sides in the general formula for medians:
where the second equation in each case reflects the automedian feature
From this can be seen the similarity relationships
There is a primitive integer-sided automedian triangle that is not generated from a right triangle: namely, the equilateral triangle with sides of unit length.
There are 18 primitive integer automedian triangles, shown here as triples of sides (a, b, c), with b ≤ 200:
| (1, 1, 1) | (13, 17, 7) | (17, 23, 7) | (25, 31, 17) | (37, 47, 23) | (41, 49, 31) |
| (61, 71, 49) | (65, 79, 47) | (85, 97, 71) | (85, 113, 41) | (89, 119, 41) | (101, 119, 79) |
| (113, 127, 97) | (125, 161, 73) | (145, 161, 127) | (145, 167, 119) | (149, 191, 89) | (181, 199, 161) |
For example, (26, 34, 14) is not a primitive automedian triple, as it is a multiple of (13, 17, 7) and does not appear above.
If is the area of the automedian triangle, by Heron's formula [3]
The Euler line of an automedian triangle is perpendicular to the median to side a. [2]
If the medians of an automedian triangle are extended to the circumcircle of the triangle, then the three points LMN where the extended medians meet the circumcircle form an isosceles triangle. The triangles for which this second triangle LMN is isosceles are exactly the triangles that are themselves either isosceles or automedian. This property of automedian triangles stands in contrast to the Steiner–Lehmus theorem, according to which the only triangles two of whose angle bisectors have equal length are the isosceles triangles. [2]
Additionally, suppose that ABC is an automedian triangle, in which vertex A stands opposite the side a. Let G be the point where the three medians of ABC intersect, and let AL be one of the extended medians of ABC, with L lying on the circumcircle of ABC. Then BGCL is a parallelogram, the two triangles BGL and CLG into which it may be subdivided are both similar to ABC, G is the midpoint of AL, and the Euler line of the triangle is the perpendicular bisector of AL. [2]
When generating a primitive automedian triangle from a primitive Pythagorean triple using the Euclidean parameters m, n, then and it follows that . As non-primitive automedian triangles are multiples of their primitives the inequalities of the sides apply to all integer automedian triangles. Equality occurs only for trivial equilateral triangles. Furthermore, because is always odd, all the sides a, b, c have to be odd. This fact allows automedian triples to have sides and perimeter of prime numbers only. For example, (13, 17, 7) has perimeter 37.
Because in a primitive automedian triangle side a is the sum of two squares and equal to the hypotenuse of the generating primitive Pythagorean triple, it is divisible only by primes congruent to 1 (mod 4). Consequently, a must be congruent to 1 (mod 4).
Similarly, because the sides are related by , each of the sides b and c in the primitive automedian is the difference between twice a square and a square. They are also the sum and difference of the legs of a primitive Pythagorean triple. This constrains b and c to be divisible only by primes congruent to ±1 (mod 8). Consequently, b and c must be congruent to ±1 (mod 8). [4]
The study of integer squares in arithmetic progression has a long history stretching back to Diophantus and Fibonacci; it is closely connected with congrua, which are the numbers that can be the differences of the squares in such a progression. [1] However, the connection between this problem and automedian triangles is much more recent. The problem of characterizing automedian triangles was posed in the late 19th century in the Educational Times (in French) by Joseph Jean Baptiste Neuberg, and solved there with the formula 2a2 = b2 + c2 by William John Greenstreet. [5]
Apart from the trivial cases of equilateral triangles, the triangle with side lengths 17, 13, and 7 is the smallest (by area or perimeter) automedian triangle with integer side lengths. [2]
There is only one automedian right triangle, the triangle with side lengths proportional to 1, √2, and √3. [2] This triangle is the second triangle in the spiral of Theodorus. It is the only right triangle in which two of the medians are perpendicular to each other. [2]
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.
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 .
A right triangle or right-angled triangle (British), or more formally an orthogonal triangle, is a triangle in which one angle is a right angle. The relation between the sides and other angles of the right triangle is the basis for trigonometry.
In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
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 Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
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.
The square root of 2 is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as or , and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.
In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense.
In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.
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 number theory, a congruum is the difference between successive square numbers in an arithmetic progression of three squares. That is, if x2, y2, and z2 are three square numbers that are equally spaced apart from each other, then the spacing between them, z2 − y2 = y2 − x2, is called a congruum.
In geometry, the spiral of Theodorus is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. However, other definitions of the term "rational triangle" also exist: In 1914 Carmichael used the term in the sense that we today use the term Heronian triangle; Somos uses it to refer to triangles whose ratios of sides are rational; Conway and Guy define a rational triangle as one with rational sides and rational angles measured in degrees—in which case the only rational triangle is the rational-sided equilateral triangle.
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one of which was stated in 1225 by Fibonacci. In its geometric forms, it states:
In Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal, but they don't have to be.
In Euclidean geometry, a right kite is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.
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 geometry, two triangles are said to be 5-Con or almost congruent if they are not congruent triangles but they are similar triangles and share two side lengths. The 5-Con triangles are important examples for understanding the solution of triangles. Indeed, knowing three angles and two sides is not enough to determine a triangle up to congruence. A triangle is said to be 5-Con capable if there is another triangle which is almost congruent to it.
