More triangle inequalities

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

LetPbe a point inside a triangleABC. Letx,y,zbe the distances fromPto the vertices and letp,q,r, be the distances to the sides. Then Erds' 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 Erdos Inequality and Other Inequalities for a Triangle. The American Mathematical Monthly. Vol. 68, No. 3 (Mar., 1961), pp. 226-230

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