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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 …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let the circumcenter of triangle

ABC

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

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: 2
config / split: / test

Official source

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