Euler's theorem in geometry

Euler's Theorem In Geometry

Euler's theorem in geometry

to get instant updates about 'Euler's Theorem In Geometry' on your MyPage. Meet other similar minded people. Its Free!


All Updates

In geometry, Euler's theorem, named after Leonhard Euler, states that the distance d between the circumcentre and incentre of a triangle can be expressed as

<math> d^2=R (R-2r) ,</math>

where R and r denote the circumradius and inradius respectively (the radii of the above two circles).

From the theorem follows the Euler inequality:
<math>R ge 2r. </math>


Let 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 mid-point 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, then 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

angle BIL = angle A / 2 + angle ABC / 2,

angle IBL = angle ABC / 2 + angle CBL = angle ABC / 2 + angle A / 2,

therefore angle BIL = angle 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&nbsp;+&nbsp;d)(R&nbsp;&minus;&nbsp;d) = 2Rr, i.e. d<sup>2</sup> = R(R&nbsp;&minus;&nbsp;2r).

External links


Read More

No feeds found

Posting your question. Please wait!...

No updates available.
No messages found
Suggested Pages
Tell your friends >
about this page
 Create a new Page
for companies, colleges, celebrities or anything you like.Get updates on MyPage.
Create a new Page
 Find your friends
  Find friends on MyPage from