什么时候一个椭圆内接于一个四边形的两条或多条边的中点处的切线？
When is an ellipse inscribed in a quadrilateral tangent at the midpoint of two or more sides ?
摘要：
在Clifford和Lachance的“Quartic Coincidences and the奇异值分解“（数学杂志，2013年12月）中，示出了如果存在中点椭圆（内接于四边形$Q$中的椭圆，其在$Q$的所有四个边的中点处相切），则$Q$必须是一个椭圆。我们加强了这一结果表明，如果$Q$是不是一个椭圆，然后有没有椭圆内接于$Q$这是切线的三个边的中点处的$Q$。第二，只有内接椭圆与$Q$的偶数两条边的中点相切的四边形是椭圆形，或者我们称之为中点对角四边形（$Q$的对角线的交点与$Q$的至少一条对角线的中点重合）。
英文摘要：
In "Quartic Coincidences and the Singular Value Decomposition" by Clifford and Lachance, Mathematics Magazine, December, 2013, it was shown that if there is a midpoint ellipse(an ellipse inscribed in a quadrilateral, $Q$, which is tangent at the midpoints of all four sides of $Q$), then $Q$ must be a parallelogram. We strengthen this result by showing that if $Q$ is not a parallelogram, then there is no ellipse inscribed in $Q$ which is tangent at the midpoint of three sides of $Q$. Second, the only quadrilaterals which have inscribed ellipses tangent at the midpoint of even two sides of $Q$ are trapezoids or what we call a midpoint diagonal quadrilateral(the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$).
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