Automedian triangle
Encyclopedia
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.
characterizing right triangle
s 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 three squared side lengths b2, a2, and c2 should form an arithmetic progression
.
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.
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.
and Fibonacci
; it is closely connected with congruent number
s, which are the numbers that can be the differences of the squares in such a progression. 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 W. J. Greenstreet.
There is only one automedian right triangle, the triangle with side lengths 1, √2
, and √3
. 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.
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...
, an automedian triangle is a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
in which the lengths of the three medians
Median (geometry)
In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...
(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
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
to the first one.
Characterization
The side lengths of an automedian triangle satisfy a formula 2a2 = b2 + c2, analogous to the Pythagorean theoremPythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
characterizing right triangle
Right triangle
A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...
s 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 three squared side lengths b2, a2, and c2 should form an arithmetic progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...
.
Construction from right triangles
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.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
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....
and could not be used to form the sides of a triangle.
Additional properties
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 whose angle bisectors have equal length are the isosceles triangles.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
Parallelogram
In Euclidean geometry, a parallelogram is a convex 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 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.
History
The study of integer squares in arithmetic progression has a long history stretching back to DiophantusDiophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...
and Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...
; it is closely connected with congruent number
Congruent number
In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property....
s, which are the numbers that can be the differences of the squares in such a progression. 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
Joseph Jean Baptiste Neuberg
Joseph Jean Baptiste Neuberg was a Luxembourger mathematician who worked primarily in geometry.-Biography:Neuberg was born on 30 October 1840 in Luxembourg City, Luxembourg. He first studied at a local school, the Athénée de Luxembourg, then progressed to Ghent University, studying at the École...
, and solved there with the formula 2a2 = b2 + c2 by W. J. Greenstreet.
Special cases
The triangle with side lengths 17, 13, and 7 is the smallest automedian triangle with integer side lengths.There is only one automedian right triangle, the triangle with side lengths 1, √2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...
, and √3
Square root of 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property...
. This triangle is the second triangle in the Spiral of Theodorus
Spiral of Theodorus
In geometry, the spiral of Theodorus is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene.-Construction:...
. It is the only right triangle in which two of the medians are perpendicular to each other.
External links
- Automedian Triangles and Magic Squares, K. S. Brown's mathpages