Viviani's Theorem

triangle size

AH

A

≈

1.73

BH

B

≈

1.73

CH

C

≈

1.73

PA'

≈

0.30

PB'

≈

0.55

PC'

≈

0.88

PA'

+

PB'

+

PC'

≈

1.73

Let ABC be an equilateral triangle and let P be a point inside ABC. Draw perpendiculars PA', PB', and PC' from P to the sides of ABC. Because ABC is equilateral, the altitudes all have the same length; call that

d

. Then PA' + PB' + PC' =

d

.

Drag P or the slider to change the figure.

External Links

Viviani's Theorem (Wolfram MathWorld)

Permanent Citation

Jay Warendorff

"Viviani's Theorem"
http://demonstrations.wolfram.com/VivianisTheorem/
Wolfram Demonstrations Project
Published: March 7, 2011