Euler's theorem in geometry

Euler's theorem:

In geometry, Euler's theorem states that the distance d between the circumcentre and incentre of a triangle is given by[1][2]

or equivalently

where R and r denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, who published it in 1765.[3] However, the same result was published earlier by William Chapple in 1746.[4]

From the theorem follows the Euler inequality:[5][6]

which holds with equality only in the equilateral case.[7]:p. 198

ProofEdit

 
Proof of Euler's theorem in geometry

Letting O be the circumcentre of triangle ABC, and I be its incentre, the extension of AI intersects the circumcircle at L. Then L is the midpoint of arc BC. Join LO and extend it so that it intersects the circumcircle at M. From I construct a perpendicular to AB, and let D be its foot, so ID = r. It is not difficult to prove that triangle ADI is similar to triangle MBL, so ID / BL = AI / ML, i.e. ID × ML = AI × BL. Therefore 2Rr = AI × BL. Join BI. Because

BIL = ∠ A / 2 + ∠ ABC / 2,
IBL = ∠ ABC / 2 + ∠ CBL = ∠ ABC / 2 + ∠ A / 2,

we have ∠ BIL = ∠ IBL, so BL = IL, and AI × IL = 2Rr. Extend OI so that it intersects the circumcircle at P and Q; then PI × QI = AI × IL = 2Rr, so (R + d)(R − d) = 2Rr, i.e. d2 = R(R − 2r).

Stronger version of the inequalityEdit

A stronger version[7]:p. 198 is

 

where a, b, c are the sidelengths of the triangle.

Euler's theorem for the escribed circleEdit

If   and   denote respectively the radius of the escribed circle opposite to the vertex   and the distance between its centre and the centre of the circumscribed circle, then  .

Euler's inequality in absolute geometryEdit

Euler's inequality, in the form stating that, for all triangles inscribed in a given circle, the maximum of the radius of the inscribed circle is reached for the equilateral triangle and only for it, is valid in absolute geometry.[8]

See alsoEdit

ReferencesEdit

  1. ^ Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry, Dover Publ., p. 186.
  2. ^ Dunham, William (2007), The Genius of Euler: Reflections on his Life and Work, Spectrum Series, 2, Mathematical Association of America, p. 300, ISBN 9780883855584.
  3. ^ Gerry Leversha, G. C. Smith: Euler and Triangle Geometry. In: The Mathematical Gazette, Vol. 91, No. 522, Nov., 2007, S. 436–452 (JSTOR 40378417)
  4. ^ Chapple, William (1746), "An essay on the properties of triangles inscribed in and circumscribed about two given circles", Miscellanea Curiosa Mathematica, 4: 117–124. The formula for the distance is near the bottom of p.123.
  5. ^ Alsina, Claudi; Nelsen, Roger (2009), When Less is More: Visualizing Basic Inequalities, Dolciani Mathematical Expositions, 36, Mathematical Association of America, p. 56, ISBN 9780883853429.
  6. ^ Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 124, ISBN 9781848165250.
  7. ^ a b Svrtan, Dragutin; Veljan, Darko (2012), "Non-Euclidean versions of some classical triangle inequalities", Forum Geometricorum, 12: 197–209.
  8. ^ Pambuccian, Victor; Schacht, Celia (2018), "Euler's inequality in absolute geoemtry", Journal of Geometry, 109 (Art. 8): 1–11, doi:10.1007/s00022-018-0414-6.

External linksEdit