Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given is a triangle on points x,y,z in the plane. This triangle has two points a and b on different sides. I would like to show that the following inequality has to hold:

$\max \{d(b,x), d(b,y), d(b,z)\} + \max \{d(a,x), d(a,y), d(a,z)\} - d(b,a) \geq \min \{d(x,y), d(x,z), d(y,z)\}$

where d(u,v) denotes the euclidean distance between u and v. I actually expect the above statement to be true even if a and b are two arbitrary points outside of the triangle.

Does anybody have an idea how to approach this?

share|cite|improve this question
What qualifies as "opposite sides" on a triangle? Each side is adjacent to the other 2. – Mike Oct 26 '12 at 22:48
@Mike I replaced the word "opposite" with "different" – andy Oct 26 '12 at 23:05
up vote 1 down vote accepted

Actually, it is true for all points inside the triangle (including the boundary). Assume that the line $ab$ intersects the side $xz$ when extended beyond $a$ and $yz$ when extended beyond $b$. Consider the quadrilateral $xaby$. We have $|xy|+|ab|\le|xb|+|ay|$ (the sum of opposite sides is not greater than the sum of diagonals). But the left hand side dominates $|ab|+\min(|xy|,|yz|,|xz|)$ while the right hand side is dominated by $\max(|ax|,|ay|,|az|)+\max(|bx|,|by|,|bz|)$. Outside the triangle the inequality may fail. Take any line $L$ and put $a$ and $b$ far away on that line on the opposite sides of the triangle. Then the difference on the left is almost the length of the projection of the triangle to $L$ but that dominates only the shortest altitude, not the shortest side.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.