最近在 Berkeley Problems in Mathematics 中看到一道题 (Problem 1.1.12) 有一个挺有意思的解法，在此记录之。
Problem: Find the maximum area of all triangles that can be inscribed in an ellipse with semiaxes $a$ and $b$ ,and describe the triangles that have maximum area.
问题：找出半轴长为 $a$ 和 $b$ 的椭圆内接三角形面积的最大值，并且描述该三角形。
一般的做法就是设出三角形三个顶点坐标，根据坐标面积公式求出最大值。但是答案给出了另一种解法，感觉十分巧妙：
Solution: Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2,(x,y)\mapsto (ax,by)$. By a well know lemma in the proof of the Change of Variable for Integration (see [1], p.524, Lemma 1), or the theorem itself when proven in its whole generality,
\begin{equation*}
\mathrm{vol} (f(T))=|det f|\cdot \mathrm{vol} A
\end{equation*}
since the determinant of $f$ is constant, following on the steps of the previous proof, the maximum for the area is achieved over the image pf an equilateral triangle and it is equal to
\begin{equation*}
\mathrm{vol}(f(T))=ab\cdot \mathrm{vol}(T)=3ab\frac{\sqrt{3}}{4}.
\end{equation*}
[1] J.E.Marsden and M.J.Hoffman. Elementary Classical Analysis. W.H.Freeman, New York, 1993.
