Geometry Problem. Post your solution in the comment box below.

Level: Mathematics Education, High School, Honors Geometry, College.

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## Sunday, October 7, 2018

### Geometry Problem 1393: Monge's Circle Theorem, Three Circles and Three Pair of Common Tangents, Collinearity

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AD / BD = rA / rB

ReplyDeleteCE / AE = rC / rA

BF / CF = rB / rC

Multiplying the 3 equations RHS = 1

Hence applying Menelaus the result follows

Sumith Peiris

Moratuwa

Sri Lanka

Let radius of circles centered A, B and C are r1, r2 and r3

ReplyDeleteConnect DAB, ECA and FCB

We have DB/DA= r2/r1

EA/EC= r1/r3

FC/FB= r3/r2

Multiply above expressions side by side we have DB/DA x EA/EC xFC/FA= 1

So D, E, F are collinear per Menelaus theorem