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 1 of 4: Introduce second circle intersections

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In plain words

Introduce second circle intersections

AU=BX,\quad AU=CY

Detailed analysis

Extend $BV$ to meet the circumcircle again at $X$ , and extend $CW$ to meet it again at $Y$ . Since $V$ lies on the perpendicular bisector of $AB$ and $W$ on that of $AC$ , symmetry about the circumcenter gives $AU=BX$ and $AU=CY$ .