化長方為方、化三角為方

GRE前夕，不寫太詳細的Blog文。今天從鄭院長的網站見到兩條關於圓規直尺作圖的問題，在這裏以「高考純數卷」形式講講吧。中四或以上的學生已經有足夠的幾何常識，可嘗試逐part完成。

a) You are given a unit length (a line segment of length 1) and two segments with lengths x and y respectively. By considering two similar triangles, show that one can construct a segment with length xy.

b) You are given a unit length and a segment with length x. By considering a circle with diameter x+1, show that one can construct a segment with length .

c) "Quadrature" a plane figure means to construct a square where its area is equal to that of the plane figure, with use of straight edge and compasses only.

i) You are given a rectangle with sides x and y respectively. Show that you can quadrature the rectangle.

ii) You are given an arbitrary triangle. Show that you can quadrature the triangle. (Hint: by i), you should already know how to quadrature any given parallelogram.)

順帶一提，quadrature a given circle（即中文的「化圓為方」）是不可能做到的。