Euler's theorem in geometry

In geometry, Euler's theorem states that the distance d between the circumcentre and incentre of a triangle can be expressed as^[1][2][3][4]

$d^2=R (R-2r) \,$

or equivalently

$\frac{1}{R-d} + \frac{1}{R+d} = \frac{1}{r},$

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 1767.^[5] However, the same result was published earlier by William Chapple in 1746.^[6]

From the theorem follows the Euler inequality:^[2][3]

$R \ge 2r,$

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

Proof

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 inequality

A stronger version^{[7]:p. 198} is

$\frac{R}{r} \geq \frac{abc+a^3+b^3+c^3}{2abc} \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}-1 \geq \frac{2}{3} \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \geq 2.$

See also

• Fuss' theorem for the relation among the same three variables in bicentric quadrilaterals
• Poncelet's closure theorem, showing that there is an infinity of triangles with the same R, r, and d
• List of triangle inequalities

References

1. Johnson, Roger A. (2007) [1929], Advanced Euclidean Geometry, Dover Publ., p. 186.
2. Alsina, Claudi; Nelsen, Roger (2009), When Less is More: Visualizing Basic Inequalities, Dolciani Mathematical Expositions 36, Mathematical Association of America, p. 56, ISBN 9780883853429.
3. Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 124, ISBN 9781848165250.
4. Dunham, William (2007), The Genius of Euler: Reflections on his Life and Work, Spectrum Series 2, Mathematical Association of America, p. 300, ISBN 9780883855584.
5. Euler, Leonhard (1767), "Solutio facilis problematum quorumdam geometricorum difficillimorum" (PDF), Novi Commentarii academiae scientiarum Petropolitanae (in Latin) 11: 103–123.
6. 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.
7. Svrtan, Dragutin; Veljan, Darko (2012), "Non-Euclidean versions of some classical triangle inequalities", Forum Geometricorum 12: 197–209.

External links

• Weisstein, Eric W., "Euler Triangle Formula", MathWorld.