MathDB

equilateral triangle

Source: 18th Austrian-Polish 1995

February 3, 2007

geometrycircumcirclegeometry proposed

Problem Statement

ABC

is an equilateral triangle.

A_{1}, B_{1}, C_{1}

are the midpoints of

BC, CA, AB

respectively.

p

is an arbitrary line through

A_{1}

.

q

and

r

are lines parallel to

p

through

B_{1}

and

C_{1}

respectively.

C_{1}A_{1}

meets the line

B_{1}C_{1}

at

A_{2}

. Similarly,

A_{1}B_{1}

meets

C_{1}A_{1}

at

B_{2}

, and

r

meets

A_{1}B_{1}

at

C_{2}

. Show that the lines

AA_{2}, BB_{2}, CC_{2}

meet at some point

X

, and that

X

lies on the circumcircle of

ABC

.

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