WOLFRAM|DEMONSTRATIONS PROJECT

Altitudes and Incircles

​
r
(AA'B)
≈
3.08
r
(AA'C)
≈
2.77
r
(BB'A)
≈
2.44
r
(BB'C)
≈
3.14
r
(CC'B)
≈
3.32
r
(CC'A)
≈
2.32
sp(ABC)
≈
23.61
AA'
≈
12.68
BB'
≈
14.35
CC'
≈
13.65
AA'
+
BB'
+
CC'
≈
40.68
r
(AA'B)
+
r
(AA'C)
+
r
(BB'A)
+
r
(BB'C)
+
r
(CC'B)
+
r
(CC'A)
+
sp(ABC)
≈
40.68
Let ABC be an acute triangle with altitudes AA', BB', and CC'. The sum of the altitudes is equal to the sum of the semiperimeter
sp(ABC)
and the radii
r(XYZ)
of the six incircles of the triangles XYZ formed by the altitudes and the sides:
AA'+BB'+CC'=sp(ABC)+r(AA'B)+r(AA'C)+r(BB'A)+r(BB'C)+r(CC'B)+r(CC'A)
.