Online Geometry theorems, problems, solutions, and related topics.
Geometry Problem. Post your solution in the comments box below.Level: Mathematics Education, High School, Honors Geometry, College.Click the figure below to view more details of problem 1169.
http://s18.postimg.org/sxwxdkf3d/pro_1169.pngDraw circumcircles of triangles ADE, ABF, BEC and CDFPer the result of problem 547 all these 4 circumcircles will concur at a point G ( see sketch)From G make perpendicular projections to AF, BF ,ED and AE at M, N, L, P.Observe that MNL, NLP, MNP and MLP are Simson lines of G to circumcircles of triangles DCF, EBC, ABF and AED.So M, N, L and P are collinear.Perform homothety transformation center G with scale factor= 2 , MNLP will become M’N’L’P’. M’N’L’ will be Steiner line of triangle CDF from point G . This line will pass through orthocenter Hc of triangle CDF ( property of Steiner line)Similarly for other 3 triangles , we will have Steiner lines N’L’P’, M’N’P’ and M’L’P’ from G of triangle BEC, ABF and AED.M’N’L’P’ will pass through Orthocenters Ha, Hb, Hc, and Hd => Ha, Hb, Hc and Hd are collinear