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A, P, Q, O are concyclic if BP : PQ : QC = b : a : c

Source: brazilian tst 01

March 16, 2005

geometrycircumcirclegeometric transformationreflectiongeometry solved

Problem Statement

Let

ABC

be a triangle with circumcenter

O

. Let

P

and

Q

be points on the segments

AB

and

AC

, respectively, such that

BP : PQ : QC = AC : CB : BA

. Prove that the points

A

,

AC

,

Q

and

O

lie on one circle. Alternative formulation. Let

O

be the center of the circumcircle of a triangle

ABC

. If

P

and

Q

are points on the sides

AB

and

AC

, respectively, satisfying

\frac{BP}{PQ}=\frac{CA}{BC}

and

\frac{CQ}{PQ}=\frac{AB}{BC}

, then show that the points

A

,

P

,

Q

and

O

lie on one circle.

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