MathDB

Let

C_1

and

C_2

be two different circles , and let their radii be

r_1

and

r_2

, the two circles are passing through the two points

A

and

B

(i)Let

P_1

C_1

and

P_2

C_2

such that the line

P_1P_2

passes through

A

. Prove that

P_1B \cdot r_2 = P_2B \cdot r_1

(ii)Let

DEF

be a triangle that it's inscribed in

C_1

, and let

D'E'F'

be a triangle that is inscribed in

C_2

. The lines

EE'

DD'

and

FF'

all pass through

A

. Prove that the triangles

DEF

and

D'E'F'

are similar
(iii)The circle

C_3

also passes through

A

and

B

. Let

l

be a line that passes through

A

and cuts circles

C_i

M_i

with

i = 1,2,3

. Prove that the value of

\frac{M_1M_2}{M_1M_3}

is constant regardless of the position of

l

Provided that

l

is different from

AB

Problem Statement