1976 AMC 12 第 16 题

Drop altitudes from $C$ and $F$; each altitude bisects the base of its isosceles triangle

Compare a half of $ABC$ with a half of $DEF$ as right triangles

Let the equal sides have length $s,$ let $AB=2a,$ and let the altitude from $F$ be $a.$ In the isosceles triangles, the altitude from $C$ bisects $AB,$ while the altitude from $F$ bisects $DE.$ A half of $ABC$ has hypotenuse $s$ and leg $a;$ a half of $DEF$ has the same hypotenuse and altitude leg $a.$ The two right triangles are congruent with their legs interchanged. Thus their relevant acute angles are complementary, so the full vertex angles $\angle ACB$ and $\angle DFE$ sum to $180^\circ.$ The corresponding half-areas are equal as well, hence the full triangle areas are equal. Statements $\mathrm{II}$ and $\mathrm{III}$ hold.

Therefore, the correct answer is E.