Indeed, the effect was good. The child really became interested in the compass and ruler, and kept asking his father new questions. Soon, his father got stuck. Let’s think about it together, what problem is it that makes dad stumped?

I heard that ruler and compass drawing can stimulate children's interest in geometry, and this father also wanted to try it. Indeed, the effect was good. The child really became interested in the compass and ruler, and kept asking his father new questions. Soon, his father got stuck.

A seemingly easy question made my father think hard for a long time, so he had to send a message to let us talk about it. Let's think about it together, what problem is it that makes dad stumped?

As shown in the figure, two circles are tangent to each other, and a straight line is also tangent to them at the same time. If you want to make a circle that is tangent to these two circles and the straight line at the same time, you can only use a ruler and compass to draw.

After some observation, I intuitively feel that the circle to be made should be in the area between the three. The father drew two vertical auxiliary lines from the centers of the two circles to their tangent points. After drawing the auxiliary lines connecting the centers of the two circles, he was at a loss.

First of all, I want to praise the children. When I was in school, my tutor often said, "The big scientists ask questions, the little scientists are busy solving the problems, and the scientific and technological migrant workers help improve the solutions." Being able to ask good questions and point out the direction of research is really amazing. This kid hit the mark and asked the question Apollonius asked more than two thousand years ago.

The problem posed by the ancient Greek mathematician Apollonius (Apollonius' Problem) is a famous geometry problem: "Given three circles on the plane, how to use a ruler and compass to construct a sum of these three known circles are all tangent circles.

When one of the circles degenerates into a straight line, Apollonius' problem becomes the ruler and compass construction problem we need to solve today, because it involves more theorems and preparatory knowledge. Today we will tell you the drawing method first, and help maintain the glorious image of the father [呲呲]. In the following articles, we will discuss the principles.

Step 1

needs to be learned. Given the two line segments L1 and L2, draw a line. The length of the segment is √ (L1·L2), as shown below. We will use this technique later. Friends who are not sure can go to our homepage to read the previous articles.

Step 2

First, go through the two center points and do it towards the respective tangent points. The vertical line, the tangent points are D and E.

Step 3

Construct a line segment on the vertical line AD, the length of the line segment is required to be 2√(AD·DE), and the trace of the construction is the dotted line part, mark the I point. , will be used later.

Step 4

In the other direction, a line segment is also constructed, and the length of the line segment is required to be 2√(BE·DE). For the drawing trace, see the dotted line part of the constructed line segment ME, which will be used later. .

Step 5

Connect the two points MI and intersect the tangent DE at N. Draw the perpendicular line of the tangent DE through point N. The center of the tangent circle we want to make is on this line.

Step 6

Take D as the center and DN as the radius. The circle intersects AD at K. Connect KE and intersect the perpendicular line passing through N at point P.

Step7

Take the midpoint Q of the line segment PN, take Q as the center, and use QN as the radius to make a circle. The circle is connected to the circle A and the circle. B and the tangent line DE are tangent, and the drawing is completed.

Dear parents, try it with your children. Is this the case? If there are still children who want to know why the drawing can be done in this way, and how to prove it, they need to learn some expanded knowledge. , such as using Descartes’ theorem, etc.

This article is written by Xiao Liuyang Jeffery to share with you his experience in overseas stimulation. ☎ Follow "Xiao Liuyang" to get more overseas K12 education information.