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Unusual Hexagon Geo
oVlad   2
N 5 minutes ago by Double07
Source: Romania Junior TST 2025 Day 1 P4
Let $ABCDEF$ be a convex hexagon, such that the triangles $ABC$ and $DEF$ are equilateral and the diagonals $AD, BE$ and $CF$ are concurrent. Prove that $AC\parallel DF$ or $BE=AD+CF.$
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oVlad
Apr 12, 2025
Double07
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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
2312 posts
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
104 posts
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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