MathLabs

Problem 2

Given a nondegenerate triangle ABCABC, with circumcentre OO, orthocentre HH, and circumradius RR, prove that ∣OH∣<3R|OH|<3R.
Step 1 of 4: Put the circumcentre at the origin
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O=0,∣a∣=∣b∣=∣c∣=RO=0,\qquad |a|=|b|=|c|=R
Detailed analysis

Embed the plane in the complex plane, put OO at 00, and denote the vertex coordinates by a,b,ca,b,c. Since the vertices lie on the circumcircle, ∣a∣=∣b∣=∣c∣=R|a|=|b|=|c|=R.