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Omni-MATH / Let the circumcenter of triangle ABC be O. H_(A) is the projection of A onto BC. The extension of AO intersects the circumcircle of BOC at A'. The projections of A' onto AB,AC are …

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problem

Let the circumcenter of triangle ABCABC be OO. HAH_A is the projection of AA onto BCBC. The extension of AOAO intersects the circumcircle of BOCBOC at $A'$. The projections of $A'$ onto AB,ACAB, AC are D,ED,E, and OAO_A is the circumcentre of triangle DHAEDH_AE. Define HB,OB,HC,OCH_B, O_B, H_C, O_C similarly. Prove: HAOA,HBOB,HCOCH_AO_A, H_BO_B, H_CO_C are concurrent
Plain-text mathematical notation (without MathML)
Let the circumcenter of triangle ABC be O. H_(A) is the projection of A onto BC. The extension of AO intersects the circumcircle of BOC at $A'$. The projections of $A'$ onto AB,AC are D,E, and O_(A) is the circumcentre of triangle DH_(A)E. Define H_(B),O_(B),H_(C),O_(C) similarly. 
Prove: H_(A)O_(A),H_(B)O_(B),H_(C)O_(C) are concurrent
Original LaTeX notation
Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly. 
Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent

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task_id
372dfa89-570a-5d00-9ac1-4516b2e4d443
task_key
test--372dfa89-570a-5d00-9ac1-4516b2e4d443
task_revision_id
5
config / split
/ test

Official source

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