EulerRelation

1. Euler's relation

The circumradius R, the inradius r and the distance OI of the centers of circumcircle and incircle of triangle ABC satisfy the relation

Let D the other intersection point of the circumcircle with the bisector of angle C, E the diametral point of D. The distances DA = DI = DB (see Bisector.html ). It follows from the similar triangles FCI and BED,
CI/IF = ED/DB => CI*ID = CI*DB = ED*IF = r*(2R).
The product CI*ID is the power of I with respect to the circumcircle, hence is equal to R²-OI². The relation results by equating this with the previous expression.
Corollary For every triangle the in- and circum-radii satisfy the inequality:
R > 2r. The equality R=2r is valid only for the equilateral triangle.

2. Sufficiency of Euler's relation

If the radii {R, r} of two circles and the distance OI of their centers satisfy the Euler's relation, then for every point C on the circumcircle draw the tangents {CA,CB}, where {A,B} are the second intersection points of these lines with the circumcircle. Then ABC has the two circles as circum- and in-circle respectively.

In fact, if the relation is valid, then triangle CIF and the right-angled triangle with hypotenuse 2R and one side DI are similar. But last right-angled triangle is equal to EBD. Thus DB=DI and this implies that the incenter of triangle ABC coincides with I. It follows that the inradius of ABC is also equal to r as claimed.

Remark The property proved implies that if there is a triangle inscribed in a circle (a) and circumscribed on a circle (b) then there is an infinity of triangles also inscribed in the same circumcircle (a) and circumscribed about the same incircle (b). This is a special case of Poncelet's more general porism about polygons simultaneously inscribed and circumscribed in two conics (see Poncelet.html ).

3. Poristic triangles

Another aspect of the previous properties is the one of study of invariants of poristic triangles i.e. invariants of the one parameter family of all triangles simultaneously inscribed and circumscribed in two given circles. The following problems are from this context.

Problem-1 The excenters of all poristic triangles are on a circle (c). {A'',B'',C''} of poristic triangles are on a circle For example, the Nagel point X₈ of these triangles moves on a circle concentric to the circumcircle.

1. Euler's relation

2. Sufficiency of Euler's relation

3. Poristic triangles

See Also