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Varying point along an arc

Source: Philippine Mathematical Olympiad

December 31, 2017

geometrySpiral SimilarityAngle ChasingLocus problemsincentercircumcircle

Problem Statement

Circles

\mathcal{C}_1

and

\mathcal{C}_2

with centers at

\mathcal{C}_1

and

\mathcal{C}_2

respectively, intersect at two points

P

and

Q

. Points

B

and

B

are varying points on

P

and

\mathcal{C}_2

, respectively, such that

P

,

Q

and

B

are collinear and

B

is always between

P

and

Q

. Let lines

S

and

QC_2

intersect at

S

, let

A

be the incenter of

C_1

, and let

S

be the circumcenter of

\Delta PIQ

. Show that as

P

and

Q

vary,

S

traces the arc of a circle whose center is concyclic with

A

,

C_1

and

C_2

.

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