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MZ is perp to BC iff CA=CB or Z ≡ O

Source: Greek Mathematical Olympiad 2011 - P4

March 2, 2011
geometrycircumcirclegeometry proposed

Problem Statement

We consider an acute angled triangle ABCABC (with AB<ACAB<AC) and its circumcircle c(O,R)c(O,R) (with center OO and semidiametre RR).The altitude ADAD cuts the circumcircle at the point EE ,while the perpedicular bisector (m)(m) of the segment ABAB,cuts ADAD at the point LL.BLBL cuts ACAC at the point MM and the circumcircle c(O,R)c(O,R) at the point NN.Finally ENEN cuts the perpedicular bisector (m)(m) at the point ZZ.Prove that: MZ⊥BC  ⟺  (CA=CB    or    Z≡O) MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right)
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