The Erdös-Mordell Inequality

PA

+

PB

+

PC

≈

28.21

2

(

PA'

+

PB'

+

PC'

)

≈

28.10

For any triangle ABC take a point P inside ABC. Let

S

1

=PA+PB+PC

be the sum of the distances from P to the vertices and let

S

2

=PA'+PB'+PC'

be the sum of the distances from P to the sides of ABC. Then

S

1

≥2

S

2

, with equality holding if and only if ABC is equilateral and P is its centroid.