tag:blogger.com,1999:blog-6933544261975483399.post5527835470938969236..comments2023-09-28T06:48:50.134-07:00Comments on Go Geometry (Problem Solutions): Geometry: Nine-Point Circle, Euler Line, Instagram, Mobile AppsAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-6933544261975483399.post-74528873256110393112016-09-23T12:06:40.167-07:002016-09-23T12:06:40.167-07:00In the triangle ABC, we know the centroid G divide...In the triangle ABC, we know the centroid G divides Orthocenter H and Circumcenter O in the ratio 2:1<br />The Medal triangle MaMbMc is similar to ABC but its centroid remains the same i.e G<br />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 <br />Hence O is the projection of H along the lines passing through the centroid. <br /><br />So in the Medal triangle, O is the orthocenter and N is the circumcenter<br />Hence OG = 2NG => If NG = 1 => OG = 2 => NO = NG+GO = 3<br />But GH = 2OG => GH = NG + NH = 2*2 = 4 => NH = 4-1=3<br /><br />Hence center of the Nine point circle of triangle ABC is the midpoint of the Euler line Anonymousnoreply@blogger.com