Orbit of planet have to be a conic section

[sorce]

How we can prove (using geomtry and not differential equation) that the orbit of planet have to be a conic section.

 

[Naty1]

I don't think you can "prove" such a thing...except maybe by observations.

Please take that as an opinion, not a firm statement, as I have been away from such things a looooong time...I surprised I even remembed there was an example of a the derivation from a textbook...


My old version of Halliday and Resnick has a derivation closely linked to your query, but not so general...for CIRCULAR orbits of two bodies m and M under the influence of each others gravitational attraction.

It starts with a center of mass of the combined bodies such that mr = MR, where r,R are the radial distances of bdoies m,M, from the center of mass, and noting the centripetal forces of the two bodies equal and figuring the gravitational attraction equal results in

GMm/(R+r)² = mw²r...

 

[sorce]

i saw in wikipedia that Newton prove it usinge geometry in his book "principia"
but i can't find download of the book/