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Similarity with centers of two circles

Source: Mexico National Olympiad Mock Exam 2019 P1

October 15, 2019

similarityCentergeometry

Problem Statement

Let

C_1

and

C_2

be two circles with centers

O_1

and

O_2

, respectively, intersecting at

A

and

B

. Let

l_1

be the line tangent to

C_1

passing trough

A

, and

l_2

the line tangent to

C_2

passing through

B

. Suppose that

l_1

and

l_2

intersect at

P

and

l_1

intersects

C_2

again at

Q

. Show that

PO_1B

and

PO_2Q

are similar triangles.
Proposed by Pablo Valeriano

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