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Orthoincentre mixup in rmo mock
Project_Donkey_into_M4   2
N an hour ago by Physicsknight
Source: Mock RMO 2018,TDP and Kayak P5
Let $\Delta ABC$ be a triangle with circumcircle $\omega$, $P_A, P_B, P_C$ be the foot of altitudes from $A, B, C$ onto the opposite sides respectively and $H$ the orthocentre. Reflect $H$ across the line $BC$ to obtain $Q$. Suppose there exists points $I,J \in \omega$ such that $P_A$ is the incentre of $\Delta QIJ$. If $M$ and $N$ be the midpoints of $\overline{P_AP_B}$ and $\overline{P_AP_C}$ respectively, then show that $I,J,M,N$ are collinear.
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Project_Donkey_into_M4
Yesterday at 6:26 PM
Physicsknight
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ANGLE EQUAL
PrimeSol   7
N Aug 3, 2016 by jayme
O -circumcenter, H - ortocenter and (OD) perpendicular (DE).
Prove that angle(DHE)=angle(ABC).
7 replies
PrimeSol
Nov 26, 2015
jayme
Aug 3, 2016
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ANGLE EQUAL
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PrimeSol
175 posts
PrimeSol
#1 h  Nov 26, 2015, 7:23 AM • 1 Y
Y by Adventure10
O -circumcenter, H - ortocenter and (OD) perpendicular (DE).
Prove that angle(DHE)=angle(ABC).
Attachments:
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kormidscoler
64 posts
kormidscoler
#2 h  Nov 26, 2015, 8:30 AM • 2 Y
Y by Adventure10, Mango247
BUTTERFLY THEOREM
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jayme
9777 posts
jayme
#3 h  Nov 26, 2015, 11:36 AM • 1 Y
Y by Adventure10
Dear Mathlinkers,
very nice idea...

Sincerely
Jean-Louis
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PrimeSol
175 posts
PrimeSol
#4 h  Nov 27, 2015, 4:03 AM • 2 Y
Y by Adventure10, Mango247
kormidscoler ?
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Gryphos
1702 posts
Gryphos
#5 h  Nov 27, 2015, 6:57 AM • 2 Y
Y by PrimeSol, Adventure10
Slightly more detailed solution with Butterfly theorem
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jayme
9777 posts
jayme
#6 h  Nov 27, 2015, 7:17 AM • 2 Y
Y by PrimeSol, Adventure10
Dear Mathlinkers,
if you like butterflies, you can see

http://jl.ayme.pagesperso-orange.fr/Docs/Papillon.pdf

Sincerely
Jean-Louis
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armpist
527 posts
armpist
#7 h  Nov 28, 2015, 3:36 AM • 2 Y
Y by Adventure10, Mango247
Dear MLs,



M.T.
This post has been edited 2 times. Last edited by armpist, Mar 24, 2017, 7:29 PM
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jayme
9777 posts
jayme
#8 h  Aug 3, 2016, 1:46 PM • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,
just an interesting link with

http://www.artofproblemsolving.com/community/c6t48f6h1283140_two_perpendiculars

Sincerely
Jean-Louis
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