InscribedTria_In

Triangle inscribed in triangle

Triangle DEF inscribed in ABC. Measuring angles at O, common intersection point of the circumcircles of triangles AEF, DEB, CDF :
angle( BOC ) = angle(A)+angle(D),
angle( AOC ) = angle(B)+angle(F),
angle( AOB ) = angle(C)+angle(E).
Thus for fixed triangles ABC and DEF (later modulo similarity) the point O is characterized by its property to view the sides of ABC under the above fixed angles.

Because of the cyclic quadrilaterals OEBD, ODCF , etc. , the segments OD, OE, OF form equal angles with the corresponding sides of the triangle.
On the story of "turning" DEF about O, look at InscribedTria_In_Tria3.html .

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