MathLabs

Problem 4

For three points A,B,CA,B,C in the plane, let m(ABC)m(ABC) be the smallest length of the three heights of triangle ABCABC, and set m(ABC)=0m(ABC)=0 when the points are collinear. Given A,B,CA,B,C, prove that for every point XX in the plane, m(ABC)≤m(ABX)+m(AXC)+m(XBC)m(ABC)\le m(ABX)+m(AXC)+m(XBC).
Step 1 of 5: Express the minimum height
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In plain words

The smallest altitude is opposite the longest side.

m(ABC)=2[ABC]am(ABC)=\frac{2[ABC]}{a}
Detailed analysis

Relabel so a=BC is the longest side, with a at least b and c. The height from A is then the smallest, and its length is twice the area divided by a.