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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
.
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