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1 area = 2025 points
giangtruong13   0
34 minutes ago
In a plane give a set $H$ that has 8097 distinct points with area of a triangle that has 3 points belong to $H$ all $ \leq 1$. Prove that there exists a triangle $G$ that has the area $\leq 1 $ contains at least 2025 points that belong to $H$( each of that 2025 points can be inside the triangle or lie on the edge of triangle $G$)X
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giangtruong13
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Nesting of cevian triangles
v_Enhance   3
N May 8, 2024 by Martin2001
Source: Subproblem of TSTST 2016/6, completely projective
Let $ABC$ be a triangle and let $DEF$ be the cevian triangle w.r.t. any point $G$. Let $\triangle XYZ$ be the cevian triangle of $\triangle DEF$ w.r.t. any point $H$. Lines $XY$ and $AB$ meet at $V$ while lines $XZ$ and $AC$ meet at $W$. Prove that points $X$, $G$ and $T = \overline{BW} \cap \overline{CV}$ are collinear.

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v_Enhance
Jul 1, 2016
Martin2001
May 8, 2024
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Nesting of cevian triangles
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Source: Subproblem of TSTST 2016/6, completely projective
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v_Enhance
6870 posts
v_Enhance
#1 h  Jul 1, 2016, 2:06 PM • 4 Y
Y by baopbc, v4913, Adventure10, Mango247
Let $ABC$ be a triangle and let $DEF$ be the cevian triangle w.r.t. any point $G$. Let $\triangle XYZ$ be the cevian triangle of $\triangle DEF$ w.r.t. any point $H$. Lines $XY$ and $AB$ meet at $V$ while lines $XZ$ and $AC$ meet at $W$. Prove that points $X$, $G$ and $T = \overline{BW} \cap \overline{CV}$ are collinear.

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DVDthe1st
341 posts
DVDthe1st
#2 h  Jul 1, 2016, 4:40 PM • 3 Y
Y by v_Enhance, Adventure10, Mango247
Just one more line: then $\triangle BZW$ and $\triangle CYV$ are perspective, hence $T,X,BZ\cap CY$ are collinear.

Hopefully that helps to make a shorter proof for TSTST/6. I'm curious to see whether the cevian nest step admits a projective approach.
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Martin2001
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Martin2001
#3 h  May 8, 2024, 3:09 PM
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How is $\triangle BZW$ and $\triangle CYV$ perspective again?
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Martin2001
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Martin2001
#4 h  May 8, 2024, 7:06 PM
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Nevermind, we already got that $\overline{BC} \cap \overline{ZY} \cap \overline{VW}$ is concurrent because $\triangle BYV$ and $\triangle CZW$ are perspective.
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