MHB Proving the Similarity of Two Acute Triangles with Perpendicular Lines

[Albert1]

Acute triangle $ABC$,3 points $D,E,F $ are on $\overline{BC},\overline{AC},\overline{AB}$
respectively ,
if $\overline{AD}\perp \overline{BC} ,\overline{DE}\perp \overline {AC} $ and $\overline{DF}\perp \overline {AB}$
prove :
(1)$\triangle ABC \sim \triangle AEF$
(2) $\overline{AO}\perp \overline {EF} $
(hrere $O$ is the circumcenter of $\triangle ABC$)

 

[Albert1]

Acute triangle $ABC$,3 points $D,E,F $ are on $\overline{BC},\overline{AC},\overline{AB}$
respectively ,
if $\overline{AD}\perp \overline{BC} ,\overline{DE}\perp \overline {AC} $ and $\overline{DF}\perp \overline {AB}$
prove :
(1)$\triangle ABC \sim \triangle AEF$
(2) $\overline{AO}\perp \overline {EF} $
(hrere $O$ is the circumcenter of $\triangle ABC$)

hint:

Spoiler

(1) prove $\angle AEF=\angle B$
(2) Prove $\angle BAO+\angle AFE=90^o$

 

[Albert1]

Acute triangle $ABC$,3 points $D,E,F $ are on $\overline{BC},\overline{AC},\overline{AB}$
respectively ,
if $\overline{AD}\perp \overline{BC} ,\overline{DE}\perp \overline {AC} $ and $\overline{DF}\perp \overline {AB}$
prove :
(1)$\triangle ABC \sim \triangle AEF$
(2) $\overline{AO}\perp \overline {EF} $
(hrere $O$ is the circumcenter of $\triangle ABC$)

solution :

Spoiler

The tags of all the related angles are maked with ,hope you can figure them out

View attachment 6558