P. Yiu, Introduction to the Geometry of the Triangle, Florida Atlantic University Lecture Notes, 2001; with corrections, 2013, available at http://math.fau.edu/Yiu/Geometry.html at 2.4.3 More on reflections: (1) (Colling) The reflections of a line L in the side lines of triangle ABC are concurrent if and only if L passes through the orthocenter. In this case, the intersection is a point on the circumcircle.
I generalization this result following:
Let $ABC$ be a triangle, $H$ is the orthocenter of the triangle $ABC$. $H_a,H_b,H_c$ are projection of $H$ to $BC,CA,AB$. Let $A_1,B_1,C_1$ lie on $AH,BH,CH$ such that: $\frac{HA_1}{HH_a}=\frac{HB_1}{HH_b}=\frac{HC_1}{HH_c}=t$. Let $D$ be any point on the plain. $D_a,D_b,D_c$ are reflection of $D$ on $BC,CA,AB$. Show that: $D_aA_1,D_bB_1,D_cC_1$ are concurrent. If $t=2$ we have Colling's theorem above
Bài viết đã được chỉnh sửa nội dung bởi khongghen: 14-04-2014 - 16:20
