WOLFRAM|DEMONSTRATIONS PROJECT

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)

.