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For reference : In the figure, $F. M , G , H$ are points of tangency. What is the value of the $MC$ segment if $AF = 4, BF = 6~ and~ AM = 8$?

enter image description here

My progress:

I couldn't see almost any information... I only know that by property FH = MG

enter image description here

peta arantes
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  • I think it is fruitful to write down (it should be part of your question) the "history" of the construction: 2 circles being given, one first construct the common tangents, then line GH which intersects the circles in $B,C$, then... – Jean Marie Sep 01 '21 at 15:25
  • $\triangle ABC$ is isosceles – endgame yourgame Sep 01 '21 at 15:47
  • @JeanMarie..In the reference, the essential data is already given...The problem is exactly as in the book...I don't see the need to demonstrate the construction... – peta arantes Sep 01 '21 at 16:10
  • You cannot "demonstrate" a construction... You would need it if you use for example Geogebra for example for attempting to provide a construction to the exact scale... – Jean Marie Sep 01 '21 at 20:11
  • @JeanMarie..My drawings are almost all scaled but there are some that have little information and it is difficult even using geogebra...but I always try to use scale in the drawings – peta arantes Sep 01 '21 at 20:21

1 Answers1

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Hint:

enter image description here

Do you see why $\angle FPG = \angle HQM ~$, where $P$ and $Q$ are centers of the given circles?

What does that tell about $\triangle ABC$?

Math Lover
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    @Math..yes..$angle FPG = ,angle HQM$, perfect..$\angle BGF =\angle MCH \implies \triangle ABC(isosceles) MC = 10-8 =2$ Thank you very much – peta arantes Sep 01 '21 at 16:19