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Grade: 11
        
Suppose we have given a triangle ∆ABC with O as it's orthocentre and C'as it's circumcentre. Find the distance between them.
7 months ago

Answers : (1)

Rohit
15 Points
							

Let O the circumcenter of triangle ABC and G its centroid. Extend OG until a point P such that OG/GP=1/2. We'll prove that P is the orthocenter H.

Draw the median AA' where A' is the midpoint of BC. Triangles OGA and PGA are similar, since GP=2GO, AG=2A'G and angle  OGA'=angle PGA. Then angle OA'G =angle PGA and OA' is parallel AP. But OA' is perpendicular perp BC so AP is perpendicular to BC, that is, AP is a height of the triangle.

Repeating the same argument for the other medians proves that P lies on the three heights and therefore it must be the orthocenter H.

The ratio is $OG/GH=1/2$ since we constructed it that way.

7 months ago
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