Barrow's inequality

In geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain points on the sides of the triangle.

It is named after David Francis Barrow.

Let P be an arbitrary point inside the triangle ABC.

From P and ABC, define U, V, and W as the points where the angle bisectors of BPC, CPA, and APB intersect the sides BC, CA, AB, respectively.

Then Barrow's inequality states that[1] with equality holding only in the case of an equilateral triangle and P is the center of the triangle.

[1] Barrow's inequality can be extended to convex polygons.

For a convex polygon with vertices

denotes the secant function.

Barrow's inequality strengthens the Erdős–Mordell inequality, which has identical form except with PU, PV, and PW replaced by the three distances of P from the triangle's sides.

It is named after David Francis Barrow.

Barrow's proof of this inequality was published in 1937, as his solution to a problem posed in the American Mathematical Monthly of proving the Erdős–Mordell inequality.

[1] This result was named "Barrow's inequality" as early as 1961.

[4] A simpler proof was later given by Louis J.