AO meets the circumcircle of \triangle OBC at X, BO meets the circumcircle of \triangle OAC at Y, and CO meets the circumcircle of \triangle OAB at Z.
Show that AX + BY + CZ \geq 9R
Determine when equality occurs.
Solution
Let AX meets BC at D. And let OO' be the diameter of the circumcircle of \triangle OBC, and let M be the intersection of OO' and BC.
\frac{OX}{OO'} = \frac{OM}{OD} OX.OD = OO'.OM = OC^2 = R^2 AX = AO + OX = R + \frac{R^2}{OD} = \frac{R(R + OD)}{OD} = R\frac{AD}{OD}
If we define E as the intersection of BY and AC, and F as the intersection of CZ and AB, then similarly: BY = R \frac{BE}{OE} CZ = R \frac{CF}{OF}
So: AX + BY + CZ = R(\frac{AD}{OD} + \frac{BE}{OE} + \frac{CF}{OF}) \geq \frac{9R}{\frac{OD}{AD} + \frac{OE}{BE} + \frac{OF}{CF}} But OD / AD = A_{\triangle OBC} / A_{\triangle ABC} and so on, so: \frac{OD}{AD} + \frac{OE}{BE} + \frac{OF}{CF} = 1 AX + BY + CZ \geq 9R
and equality happens when the area of \triangle OBC is exactly \triangle ABC / 3. And this happens only when O is the centroid of \triangle ABC, which means ABC is an equilateral.
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