Inscribed Ellipse in a Parallelogram

Geometry Level 4

Parallelogram \(ABCD\) has \( AB = 20 \) and an altitude of \( 10 \), and \( \angle DAB = 60^{\circ} \). The ellipse shown is the unique ellipse that is inscribed in the parallelogram and tangent to the midpoints of its four sides. Find the sum of the lengths of the major and minor axes and the angle (in degrees) that the major axis makes with \(AC\).

That is, enter as your answer \( 2a + 2b + \phi \), where \( a \) is the length of the semi-major axis, \( b \) is the length of the semi-minor axis, and \( \phi \) is the angle between the major axis and \( AC \).


Inspiration.

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