WOLFRAM|DEMONSTRATIONS PROJECT

A Generalization of Hansen's Theorem

R	≈	2.38	AH	≈	1.22	BH	≈	3.47	CH	≈	2.27
r	≈	1.10	r A	≈	4.65	r B	≈	2.39	r C	≈	3.59

2

r

A

+

2

r

B

+

2

r

C

+

2

r

≈

41.42

2

AH

+

2

BH

+

2

CH

+

4

2

R

≈

41.42

Let ABC be a triangle and H its orthocenter. Let

r

and R be the radii of the incircle and circumcircle. Let

r

A

,

r

B

, and

r

C

be the radii of the excircles opposite A, B, and C. Then

2

r

A

+

2

r

B

+

2

r

C

+

2

r

=

2

AH

+

2

BH

+

2

CH

+4

2

R

.