MathDB

So Many Circles

Source: KöMaL A. 750

March 19, 2022

geometrykomalcircles

Problem Statement

Let

k_1,k_2,\ldots,k_5

be five circles in the lane such that

k_1

and

k_2

are externally tangent to each other at point

T,

k_3

and

k_4

are exetrnally tangent to both

k_1

and

k_2,

k_5

is externally tangent to

k_3

and

k_4

at points

U

and

V,

respectively, and

k_5

intersects

k_1

at

P

and

Q,

like shown in the figure. Prove that

\frac{PU}{QU}\cdot\frac{PV}{QV}=\frac{PT^2}{QT^2}.

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