Jack D'Aurizio Proof of Sawayama's Lemma
Through vertex A of ΔABC a straight line AD is drawn with D on BC. Let circle C1 tangent to AD at F, CD at E, and the circumcircle C2 of ΔABC at K. Then the chord EF passes through the incenter I of ΔABC.
By Casey's theorem, applied to A, B, C (as 1-point circles) and circle C1 (all "touching" C2), we have that:
AF · BC + AB · CE = AC · BE
that is equivalent to:
|(1)||a · AF + a·c = (b + c) BE,|
where a, b, c are the lengths of sides BC, AC, and AB, as usual.
Let L be the inetersection of BC with the bisector AI of angle at A. Consider now the six points A, I, L, D, E, F. In virtue of Menelaus' theorem and its converse, E, F, I are collinear if and only if in ΔADL,
(LE/DE) · (DF/AF) · (AI/IL) = 1
Which is equivalent to:
(LE/AF) · (AI/IL) = 1
AI/IL = (b + c)/a,
so, in order to prove the collinearity of I, E, F, suffice it to prove that:
|(2)||a · AF = LE · (b + c).|
Again by the angle bisector theorem,
Note: There is another poof due to Sohail Farhangi.
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