MathDB

In non-isosceles acute

{}{\triangle ABC}

AP

BQ

CR

is the height of the triangle.

A_1

is the midpoint of

BC

AA_1

intersects

QR

K

QR

intersects a straight line that crosses

{A}

and is parallel to

BC

at point

{D}

, the line connecting the midpoint of

AH

and

{K}

intersects

DA_1

A_2

. Similarly define

B_2

C_2

{}\triangle A_2B_2C_2

is known to be non-degenerate, and its circumscribed circle is

\omega

. Prove that: there are circles

\odot A'

\odot B'

\odot C'

tangent to and INSIDE

\omega

satisfying: (1)

\odot A'

is tangent to

AB

and

AC

\odot B'

is tangent to

BC

and

BA

, and

\odot C'

is tangent to

CA

and

CB

. (2)

A'

B'

C

' are different and collinear. Created by Sihui Zhang

Problem Statement