# Construction of two Focus for conic section with five points

Using an unnumbered ruler and a caliper you should Construction of two Focus for conic section with five points.

The difficulty is that the conic is not drawn, only the five points must be used

I was interested in this question in the past and I am asking it now for the sake of knowledge sharing because I think there are a lot of people who are interested in getting the build steps, I will not put the full details of the solutions because it will be too long but I will put simple signs and graphics, you can add more details in the other answers...

• math.stackexchange.com/questions/2070274/… Commented Jun 20, 2023 at 16:46
• Nice, different methods were used there Commented Jun 20, 2023 at 17:14
• See also this answer, where I discuss how to construct the axes: math.stackexchange.com/a/4720458/255730 Commented Jun 20, 2023 at 17:24
• Thank you very much Commented Jun 20, 2023 at 17:28

A conic section is uniquely defined by five distinct points on its circumference because two different conic sections cannot intersect at more than four points.

The elements of a conic section can be constructed with the knowledge of five of its points analytically, but this does not mean that it cannot be done geometrically, although the construction is difficult - and a challenge - but it can be done with the traditional tools of geometry, the ruler and the compass.

To solve the big problem, we will solve a series of step-by-step problems:

1. Create a sixth point at a random position. This is done using Pascal's theorem

2. Constructing a chord passing through a known point and parallel to a known straight line.

3. Create the center of the conical cut. This is done based on the following theorem: The midpoints of parallel strings in a conic section pass through a center

4. Create a tangent to a point. Where the tangent is considered a special feature of the hypotenuse Tangent can also be created using Pascal's theorem

5. Create two conjugated chords. This construct relies on the use of affine shunts

6. Determine the two axes of symmetry of the segments. You can see the following link

7. Create two symmetrical tangents with respect to the axis of symmetry, then create the two foci.

Finally, I apologize for the bad language as I write using Google Translate. Feel free to edit the text if you find this necessary

• Which software do you use to produce your figures ? Commented Jun 26, 2023 at 6:55
• I design the original geometric drawing using Geogebra, then I place the Geogebra drawing as a layer in a photo editor and manually recreate the elements using it. Here is the link to the application: play.google.com/store/apps/…. Commented Jun 26, 2023 at 8:25
• Thanks for the information. Commented Jun 26, 2023 at 8:57