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 3 of 5: Sum the areas
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In plain words

Area additivity converts three local estimates into the target global estimate.

[ABX]+[BCX]+[CAX]=[ABC][ABX]+[BCX]+[CAX]=[ABC]
Detailed analysis

The three triangles partition ABC, so their areas sum to [ABC]. The previous lower bound is therefore exactly m(ABC). This proves the result for X inside or on the triangle.