## Saturday, September 17, 2016

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## Saturday, September 17, 2016

Online Geometry theorems, problems, solutions, and related topics.

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In the triangle ABC, we know the centroid G divides Orthocenter H and Circumcenter O in the ratio 2:1

ReplyDeleteThe Medal triangle MaMbMc is similar to ABC but its centroid remains the same i.e G

We can see that Ma is projection of A along the lines BMb, CMc and similarly Mb is projection of B and Mc is projection of C

Hence O is the projection of H along the lines passing through the centroid.

So in the Medal triangle, O is the orthocenter and N is the circumcenter

Hence OG = 2NG => If NG = 1 => OG = 2 => NO = NG+GO = 3

But GH = 2OG => GH = NG + NH = 2*2 = 4 => NH = 4-1=3

Hence center of the Nine point circle of triangle ABC is the midpoint of the Euler line