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# RMO practice!

In a triangle $$ABC$$,with orthocentre $$H$$ and circumcentre $$O$$, a perpendicular is drawn from $$O$$ to side $$BC$$, then prove that $$AH=2OE$$.

1 year, 10 months ago

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Let $$D,E,F$$ be the midpoints of the sides $$BC$$, $$CA$$ and $$AB$$ respectively. It is easy to prove that $$O$$, the circumcenter of $$\Delta ABC$$, is the orthocenter of $$\Delta DEF$$. And now we can achieve triangle $$\Delta DEF$$ from $$\Delta ABC$$ by making a transformation about centroid, i.e. squeezing the length $$AG$$ to half and rotating $$180^0$$ gives $$DG$$. Similarly we do this for $$BG$$ and $$CG$$ to get $$EG$$ and $$FG$$. So, what happens the.... length of the line segment joining orthocenter and a vertex of triangle $$\Delta ABC$$ is twice the line joining orthocenter and a vertex of triangle $$\Delta DEF$$. But first corresponds to $$AH$$ and second corresponds to $$OD$$. Therefore, $$AH=2OD$$. · 1 year, 10 months ago