Malfatti-Steiner Problem I. A. Sakmar, University of ... - MAA Sections

Fig.8 Draw the circle through ß, D, and ß 2 . Call this circle S u .
S u intersects S t at E.
Connect C 1 to E, also ß 2 to E.
Connect C 2 to ß, also E to ß.
We will prove that E lies on the circle with the center A, which goes through
C 1 , and ß 2 .
From the circle S t ∠ C 1 Eß = 180 o – ∠ C 1 C 2 ß
From the circle S u
∠ ß 2 Eß = 180 o – ∠ ß 2 Dß
Adding
∠ C 1 Eß + ∠ ß 2 Eß = 360 o – (∠ C 1 C 2 ß + ∠ ß 2 Dß)
But ∠ C 1 Eß + ∠ ß 2 Eß = 360 o – ∠ C 1 Eß 2
Hence ∠ C 1 Eß 2 = ∠ C 1 C 2 ß + ∠ ß 2 Dß
But ∠ C 1 C 2 ß + ∠ ß 2 Dß = 360 o – ∠ A – ∠ C 2 ßD
∠ C 2 ßD is an angle in the circle S M and ∠ C 2 ßD = 180 o -
2
1 ∠ A
Finally ∠ C 1 Eß 2 = 360 o - ∠ A - 180 o + ∠
2
1 A = 180 o - ∠
2
1 A
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