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So, the task is to find a minimum distance from a given point $T$ to cone. The cone is represented with points $a,b,c$, where points $a$ and $b$ form a line that represent a symmetry axis, and points $a$ and $c$ form a line that represents slant height of the cone. All $a,b$ and $c$ are of type Point: $x,y,z \in R^3$. Point is a class with $x,y,z$ members, and Cone is a class with a,b,c members of type Point.

I found the equations of those two lines by formula and also their intersection point, which would be the vertex of the cone, but how do I find the equation of the cone? And then how will I find the distance? Could I find a line which contains point $T$ that is normal to the surface of cone, and then somehow distance?

untitled
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    I recommend the following: find the line-segment from $T$ that is perpendicular to the axis of symmetry (i.e. the shortest line from $T$ to the axis of symmetry), then use similar triangles to find the length of the component of this line segment that is normal to the slant of the cone. – Ben Grossmann Aug 14 '20 at 09:24
  • I edited the post, sorry but I still don't know how to solve it, not sure how to use similar triangles – untitled Aug 14 '20 at 12:02
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    If $a$ and $c$ form a line that represents the slant height of the cone, and $a$ is also on the axis, then surely $a$ is the vertex. Your diagram does not match the description. – David K Aug 14 '20 at 13:25
  • I must have accidentally mistaken it. I agree with you. – untitled Aug 14 '20 at 13:34

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To elaborate on my comment, here is a labeled version of your picture:

enter image description here

Note that $$ \frac{\overline{CD}}{\overline{BC}} = \frac{\overline{AO}}{\overline{BO}}. $$ With that, it suffices to compute the coordinates of points $A$ and $B$.

Ben Grossmann
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  • Careful, there are now points labeled $a$, $A$, $c$, and $C$. Also, I am not convinced that even the original diagram is drawn correctly. – David K Aug 14 '20 at 13:28