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Nordic 2025 P3
anirbanbz   8
N an hour ago by Primeniyazidayi
Source: Nordic 2025
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.
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anirbanbz
Mar 25, 2025
Primeniyazidayi
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Conic through six Circumcenters
TelvCohl   1
N Sep 23, 2015 by ancamagelqueme
Source: Own
Given a triangle $ \triangle ABC $ with circumcenter $ O $ and a point $ P $. Let $ \triangle DEF $ be the circumcevian triangle of $ P $ WRT $ \triangle ABC $. Let $ A_1, $ $ A_2 $ be the intersection of $ BC $ with $ FD, $ $ DE $, respectively (define $ B_1, $ $ B_2, $ $ C_1, $ $ C_2 $ similarly). Let $ O_1, $ $ O_2, $ $ O_3, $ $ O_4, $ $ O_5, $ $ O_6 $ be the circumcenter of $ \triangle AC_1F, $ $ \triangle BC_2F, $ $ \triangle BA_1D, $ $ \triangle CA_2D, $ $ \triangle CB_1E, $ $ \triangle AB_2E $, respectively. Let $ \mathcal{C}_T $ be the conic with center $ T $ passing through $ A_1, $ $ A_2, $ $ B_1, $ $ B_2, $ $ C_1, $ $ C_2 $. Let $ \mathcal{C}_S $ be the conic with center $ S $ passing through $ O_1, $ $ O_2, $ $ O_3, $ $ O_4, $ $ O_5, $ $ O_6 $. Let $ (T_1, T_2), $ $ (S_1, S_2) $ be the focus of $ \mathcal{C}_T, $ $ \mathcal{C}_S $, respectively. Prove that $ S $ is the midpoint of $ OT $ and $ T_1T_2 $ $ \perp $ $ S_1S_2 $.
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TelvCohl
Sep 22, 2015
ancamagelqueme
Sep 23, 2015
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Conic through six Circumcenters
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TelvCohl
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TelvCohl
#1 h  Sep 22, 2015, 1:11 PM • 4 Y
Y by mineiraojose, enhanced, Adventure10, Mango247
Given a triangle $ \triangle ABC $ with circumcenter $ O $ and a point $ P $. Let $ \triangle DEF $ be the circumcevian triangle of $ P $ WRT $ \triangle ABC $. Let $ A_1, $ $ A_2 $ be the intersection of $ BC $ with $ FD, $ $ DE $, respectively (define $ B_1, $ $ B_2, $ $ C_1, $ $ C_2 $ similarly). Let $ O_1, $ $ O_2, $ $ O_3, $ $ O_4, $ $ O_5, $ $ O_6 $ be the circumcenter of $ \triangle AC_1F, $ $ \triangle BC_2F, $ $ \triangle BA_1D, $ $ \triangle CA_2D, $ $ \triangle CB_1E, $ $ \triangle AB_2E $, respectively. Let $ \mathcal{C}_T $ be the conic with center $ T $ passing through $ A_1, $ $ A_2, $ $ B_1, $ $ B_2, $ $ C_1, $ $ C_2 $. Let $ \mathcal{C}_S $ be the conic with center $ S $ passing through $ O_1, $ $ O_2, $ $ O_3, $ $ O_4, $ $ O_5, $ $ O_6 $. Let $ (T_1, T_2), $ $ (S_1, S_2) $ be the focus of $ \mathcal{C}_T, $ $ \mathcal{C}_S $, respectively. Prove that $ S $ is the midpoint of $ OT $ and $ T_1T_2 $ $ \perp $ $ S_1S_2 $.
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ancamagelqueme
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ancamagelqueme
#2 h  Sep 23, 2015, 9:59 PM • 2 Y
Y by Adventure10, Mango247
Another property:

The circumscribed circles to triangles AFC1 and BFC2 intersect again in F'.
The circumscribed circles to triangles BFC2 and BDA1 intersect again in B'.
The circumscribed circles to triangles BDA1 and CDA2 intersect again in D'.
The circumscribed circles to triangles CDA2 and CEB1 intersect again in C'.
The circumscribed circles to triangles CEB1 and AEB2 intersect again in E'.
The circumscribed circles to triangles AEB2 and AFC1 intersect again in A'.

Then, the six points A ', B', C ', D', E 'and F' are on the same circle.
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