Problem 2

The angle at

A

is the smallest angle in triangle

ABC

. The points

B,C

divide its circumcircle into two arcs. Let

U

be an interior point of the arc

BC

not containing

A

. The perpendicular bisectors of

AB

and

AC

meet line

AU

V

and

W

, respectively. The lines

BV

and

CW

meet at

T

. Show that

AU=TB+TC

Step 2 of 4: Relate the arcs

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In plain words

Relate the arcs

\widehat{AX}=\widehat{BU},\quad\widehat{AY}=\widehat{UC}

Detailed analysis

The same perpendicular-bisector symmetry gives equality of the corresponding arcs: $\widehat{AX}=\widehat{BU}$ and $\widehat{AY}=\widehat{UC}$ . Therefore $\widehat{XY}=\widehat{BC}$ .