HMMT 二月 1998 · 几何 · 第 8 题

英文原题

Question Eight . [6 points]
It is not possible to construct a segment of length π using astraightedge, compass, and a given segment of length 1. Thefollowing construction, given in 1685 by Adam Kochansky, yields asegment whose length agrees with π to five decimal places:
Construct a circle of radius 1 and call its center O . Construct adiameter AB of this circle and a line l tangent to the circle at A .
Next, draw a circle with radius 1 centered at A , and call one of theintersections with the original circle C . Now from C draw an arc ofradius 1 intersecting the circle around A at D , where D lies outsideof the circle centered at O . Draw OD and let E be its point ofintersection with l . Construct H on AE such that A is between
H and E , and HE =3.
The distance between B and H is then close to π ; calculate its exactvalue.