俄罗斯总理不为人知的一面，竟然出了一道初中数学题

今天讲个很好玩的数学题。不必恐惧，内容均为初中知识。

每年的9月1日，是俄罗斯中学、高中以及高校秋季新学年开始的第一天，这天在俄罗斯被称为“知识日”。

俄罗斯总理米哈伊尔·米舒斯京在知识日这天，访问了一所学院。

在一个班级里，他一时兴起，出了这道题。

米舒斯京可不简单。他生于1966年，大学专业是系统工程，主修计算机辅助设计。工作以后，还读过一个经济学博士。在担任总理之前，他是俄罗斯联邦税务局长。

我们一起看下这道有意思的数学题。

Construct a perpendicular from the (red) point on the circle to the diameter, without using any measuring devices.

在不使用任何测量设备的情况下，从圆上的（红色）点画一条直径的垂直线。

In other words, given a circle with a diameter marked on it, and a point on the circle, can you find a way to draw a line from the point that hits the diameter at a right angle. (As marked in green above.)

换句话说，给定一个标有直径的圆及圆上的一个点，你能否找到一种方法，从圆上的这个点画一条线与直径成直角。（如上绿色标记所示。）

What makes this question interesting is the banning of measuring devices. Compass and marked ruler are verboten. All you are allowed is an unmarked ruler to draw straight lines.

这个问题之所以有趣，是因为禁止使用测量设备。不允许使用指南针和有标记的尺子。你只准用一把没有标记的尺子画直线。

I also gave you the following two pieces of information: 

我还提供了以下两条信息：

1) The angle subtended by a point on a circle to the two ends of a diameter is a right angle. 

1）圆上的点与直径两端的夹角是直角。

王不留（wbliu85）注：圆上任意一点与直径组成的三角形是直角三角形。

2) The altitude of a triangle is the line from a corner that meets the opposing side at a right angle. In acute triangles the three altitudes will always intersect.

2） 三角形的高是从一个角到对边成直角的直线。在锐角三角形中，三条高始终相交。

王不留（wbliu85）注：锐角三角形的三条高交于一点。

Solution

答案

STEP 1 Draw lines from the point to both ends of the diameter. Chose another point on the circle and do the same. The angles from both of these points to the ends of the diameter are right angles, as illustrated.

步骤1：从点到直径两端画线。选择圆上的另一个点，并做相同操作。从这两个点到直径两端的角度为直角，如图所示。

STEP 2 Extend the lines that go through both points until they meet, as illustrated. Consider this point of intersection as the corner of a “big” triangle whose opposite side is the diameter of the circle. Two of the altitudes of this big triangle are already drawn: these are the thin blue lines that go from each of the two ends of the diameter to the opposite sides. 

步骤2：将穿过两个点的线延伸，直到它们相交，如图所示。将此交点视为一个“大”三角形的顶点，它的对边是圆的直径。这个大三角形的两个高度已经画出来了：从直径两端到对边的蓝色细线。

We know that altitudes intersect, so when we draw a line from the top of the big triangle though the intersection of the two lines below it, that must also be an altitude, which means that it hits the diameter at a right angle.

我们知道高是相交的，（在锐角三角形中，三条高始终相交。）所以当我们从大三角形的顶部通过下面两条线的交点画一条线时，那一定也是一条高，这意味着它与直径成直角。

STEP 3 Now we have constructed a line perpendicular to the diameter, our final task is to draw a line parallel to it one that goes through the red point. To do this, draw a line from where our perpendicular touches the circle, through the red dot, to an extension of the diameter.

步骤3：现在我们已经画了一条垂直于直径的线（即，垂直线），我们的最终任务是画一条平行于这条垂直线且穿过红点的线。要做到这一点，从垂直线与圆的交点，画一条线，穿过红点，直到与直径的延伸线相交。

STEP 4 Draw a line from the other intersection of the perpendicular and the circle, and join it to the point on the diameter marked by the previous step. This line must intersect the circle at the mirror point of the red dot. The line that joins them is the perpendicular we were after.

步骤4：从垂直线和圆的另一个交点绘制一条线，并将其连接到上一步直径延伸线上的点。此线必在红点的镜像点处与圆相交，连接这个交点与红点的线就是我们要找的垂直线。

王不留（wbliu85）注：在文中 after  是介词，“追寻，追求”之意。

Ta dah! And here is Russian PM Mishustin’s drawing of the solution.

看吧！俄罗斯总理米哈伊尔·米舒斯京在解题。