Problem 3

Let

D

be an interior point of the acute triangle

ABC

with

AB>AC

so that

\angle DAB=\angle CAD

. The point

E

on the segment

AC

satisfies

\angle ADE=\angle BCD

, the point

F

on the segment

AB

satisfies

\angle FDA=\angle DBC

, and the point

X

on the line

AC

satisfies

CX=BX

. Let

O_1

and

O_2

be the circumcenters of the triangles

ADC

and

EXD

, respectively. Prove that the lines

BC

EF

, and

O_1O_2

are concurrent.

Step 6 of 6: Conclude the concurrency

P\in BC\cap EF\cap O_1O_2

Detailed analysis

By definition $P\in BC$ and $P\in EF$ , and the preceding step gives $P\in O_1O_2$ . Thus $BC$ , $EF$ , and $O_1O_2$ are concurrent at $P$ .