An Inequality for Tripolar Coordinates Related to the Brocard Angle

AM + BM + CM	≥	S ω +S 3
8.9093		8.4641

For a triangle

ABC

and a point

M

, let

S=2ABC

,

S

ϕ

=Scotϕ

and

ω

be the Brocard angle of

ABC

.

(The tripolar coordinates of

M

with respect to

ABC

are

(AM,BM,CM)

.)

Then

AM+BM+CM≥

S

ω

+S

3

;

equality holds when

ABC

is an acute triangle and

M

is its Fermat point.

External Links

The Erdös-Mordell Inequality

Erdős-Mordell Theorem (Wolfram MathWorld)

The Sum of the Trilinear Coordinates of a Point

Conway Triangle Notation

Relating Trilinear and Tripolar Coordinates for a Triangle

Brocard Angle (Wolfram MathWorld)

Permanent Citation

Minh Trinh Xuan

"An Inequality for Tripolar Coordinates Related to the Brocard Angle"
http://demonstrations.wolfram.com/AnInequalityForTripolarCoordinatesRelatedToTheBrocardAngle/
Wolfram Demonstrations Project
Published: May 24, 2022