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Euler's Theorem for Pedal Triangles

​
R
≈
6.67
OP
≈
2.57
area(ABC)
≈
55.37
area(A'B'C')
≈
11.79
1
4
|1-
2
OP
2
R
|
area(ABC)
≈
11.79
Let ABC be a triangle and P an arbitrary point. Let A', B', and C' be the feet of the perpendiculars from P to the sides of ABC. Let O and R be the center and radius of the circumcircle. Then
area(A'B'C')=
1
4
1-
2
OP
2
R
area(ABC)
.
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