MathLabs

Problem 2

Given a nondegenerate triangle ABCABC, with circumcentre OO, orthocentre HH, and circumradius RR, prove that ∣OH∣<3R|OH|<3R.
Step 2 of 4: Use the orthocentre coordinate
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h=a+b+ch=a+b+c
Detailed analysis

For three points on a circle centred at the origin, the standard complex-coordinate formula for the orthocentre is h=a+b+ch=a+b+c. Thus ∣OH∣=∣h∣=∣a+b+c∣|OH|=|h|=|a+b+c|.