Yesterday I wrote about a triangle inequality discovered by Paul Erdős.

Let P be a point inside a triangle ABC. Let x, y, z be the distances from P to the vertices and let p, q, r, be the distances to the sides. Then Erdős’ inequality says

x + y + z ≥ 2(p + q + r).

Using the same notation, here are four more triangle inequalities discovered by Oppenheim [1].

• px + qy + rz ≥ 2(qr + rp + pq)
• yz + zx + xy ≥ 4(qr + rp + pq),
• xyz ≥ 8pqr
• 1/p+ 1/q + 1/r ≥ 2(1/x + 1/y + 1/z)

[1] A. Oppenheim. The Erdös Inequality and Other Inequalities for a Triangle. The American Mathematical Monthly. Vol. 68, No. 3 (Mar., 1961), pp. 226–230

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