MathLabs

Problem 2

Given a nondegenerate triangle ABCABC, with circumcentre OO, orthocentre HH, and circumradius RR, prove that ∣OH∣<3R|OH|<3R.
Step 3 of 4: Apply the triangle inequality
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∣OH∣=∣a+b+c∣≤∣a∣+∣b∣+∣c∣=3R|OH|=|a+b+c|\le|a|+|b|+|c|=3R
Detailed analysis

The triangle inequality gives ∣OH∣=∣a+b+c∣≤∣a∣+∣b∣+∣c∣=3R|OH|=|a+b+c|\le|a|+|b|+|c|=3R.