Licia C.

asked • 04/30/20

Proving 3 points determine a unique plane

I'm trying to prove the early "Postulates" (#6-10 or so) of Euclidean geometry. I'm trying to rely on Euclid's 5 and his definitions - only. I don't want to demonstrate using some sort of manipulative. I can prove the uniqueness of the plane determined by three, given, non-colinear lines. I can't figure out how to prove that such a plane exists. Why MUST there exist a plane that contains any set of three non-colinear points?

Mark M.

Postulate by definition are not proven. They are accepted as true without proof.
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04/30/20

Licia C.

Euclid worked with 5 postulates, and his definitions. Using those 5 and the definitions, he was able to prove all of the other "Postulates" we scrub High Schoolers start with - like that 3 non-conlinear points determine a unique plane. I can prove that 3 non-conlinear points determine AT MOST one plane. I can't force a plane to exist by using Euclid's 5 plus definitions. I don't want to demonstrate (use manipulatives) or jump out to some other branch of mathematics (vectors to find the equation).
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04/30/20

1 Expert Answer

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Richard P. answered • 05/01/20

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One approach is the following. The three non-colinear points (call them A, B and C) form a triangle. For any triangle there is a circumcenter. There are several theorems that involve the circumcenter. The circumcenter is equidistant from the three corner points. The locus of points equidistant from points A and B is a plane containing the circumcenter. Similarly, the locus of points equidistant from the points B and C is a plane containing the circumcenter. And also, the locus of points equidistant from the points C and A is a plane containing the circumcenter. This is related to the well-known theorem that the perpendicular bisectors of the sides of a triangle all concur at the circumcenter. The intersection of there three planes must define a line. That line is perpendicular to a plane. The direction of that line specifies a normal vector to a plane. Obviously that plane contains the circumcenter. Thus that plane must be the plane containing the three points, A, B and C.

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Licia C.

I've started on this in order to prove that three non-colinear points determine a unique circle. That the circumcircle of a triangle is unique is a by-product of that work. In order to do the circle part, I need to know that I'm working in a specific plane - the one specified by the three, given, non-colinear plane. At this website, https://1.cdn.edl.io/onX4APDBoweM9olDEbrKK3HtcsJ1havIqBMIaJqOBSCfYdkf.pdf, they list this as postulate 2.2 Since it's NOT one of Euclid's necessary and sufficient five postulates, we should be able to prove it using just those five and the definitions. Since I can't, I'm missing something - probably a cute trick. It is math afterall. What is it about those that force the plane to exist?
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05/01/20

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