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 3 of 4: Use equal angles

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In plain words

Use equal angles

TY=TB

Detailed analysis

Equal arcs imply $\angle BYC=\angle XBY$ . Since $B,V,X,T$ are collinear and $C,W,Y,T$ are collinear, these are the base angles of triangle $TBY$ , so $TY=TB$ .