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Monday, May 4, 2015

Circumcenter and 3 circumcircles

Given an acute triangle ABC and its circumcircle is centered at O and has radius R.

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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