# Show that three point $G,H,G_1$ are collinear.

Triangle $$ABC$$ has centroid $$G$$ and orthcenter $$H$$. Line (through $$A$$) is perpendicular to $$GA$$, line (through $$B$$) is perpendicular to $$GB$$, line (through $$C$$) is perpendicular to $$GC$$ cut at three points which form a new triangle $$A_1B_1C_1$$. This new triangle has centroid $$G_1$$. Show that three point $$G,H,G_1$$ are collinear.

I have tried to so this problem with lots of theorems. But I can't find the way to solve. Or using any lemma? Help me to find and draw any auxiliary geometry element.

• There is no reason why $G=H$, let alone $G,H,G_1$ concur. Do you mean $G.H,G_1$ collinear instead? – user10354138 Dec 7 '18 at 14:38
• Oh. I'm sorry. It is "colinear". – Trong Tuan Dec 7 '18 at 15:02
• Recall the Euler line... – user10354138 Dec 7 '18 at 15:24

Taking @user10354138's comment, here's how we attack the problem: We will show that the midpoint $$O$$ of $$GG_1$$ is the circumcenter of $$\triangle ABC$$. In particular, we can actually show that $$G$$ is the midpoint of $$HG_1$$.

In the picture above, $$A_2,B_2,C_2$$ are midpoints of $$GA_1,GB_1,GC_1$$ respectively. Then $$C_2$$ is on the perpendicular bisector of $$AB$$. So it is sufficient to show that $$C_2O\perp AB$$, or $$C_1G_1\perp AB$$.

Now, if we consider a triangle $$XYZ$$ with sides (parallel to) the medians of $$\triangle ABC$$, then

1. The sides of $$\triangle XYZ$$ and $$\triangle A_1B_1C_1$$ are pairwise perpendicular.
2. The medians of $$\triangle XYZ$$ are parallel to the sides of $$\triangle ABC$$.

(The existence/construction of $$\triangle XYZ$$ and the proofs of the above statements are classical and left to you.)

From (1), it follows that $$\triangle XYZ$$ and $$\triangle A_1B_1C_1$$ are similar and their medians are pairwise perpendicular. This and (2) yield that $$C_1G_1\perp AB$$ and so on, which is what we are looking for.

• "From (1), it follows that △XYZ and △A1B1C1 are similar and their medians are pairwise perpendicular. This and (2) yield that $C_1G_1$⊥AB". Can you make it more clear? I still hav not understood. Thank you – Trong Tuan Dec 9 '18 at 13:44