a rather interesting but simple math problem

by MaxEntropy_Man Tue Feb 14, 2012 11:08 pm

A,B,C,D are four distinct points in three space. Suppose each
of the angles ABC, BCD, CDA, and DAB are right angles.
Show that all four points lie in the same plane.

(from the purdue math department's problem of the week series).

by Kris Wed Feb 15, 2012 12:14 am

If I understand the problem right, let's say point D lies outside of the plane, only one of the angles made at A or C by the line connecting to D can possibly be a right angle. Rather, angles BAD and BCD cannot both be right angles, since AD and CD will be parallel in that case.

Likewise, with any the other three points. Therefore, the 4 points must lie on the same plane, if all 4 angles are right angles.

by MaxEntropy_Man Wed Feb 15, 2012 12:33 am

kris: yes i suppose that's a euclidean geometry proof. i'll post my solution tomorrow.

by Kris Wed Feb 15, 2012 12:38 am

Ok, will look for it.

by MaxEntropy_Man Wed Feb 15, 2012 5:26 pm

not sure if this is going to work:

page 1 of my solution

page 2 of my solution

by Guest Thu Feb 16, 2012 1:09 am

MaxEntropy_Man wrote:A,B,C,D are four distinct points in three space. Suppose each
of the angles ABC, BCD, CDA, and DAB are right angles.
Show that all four points lie in the same plane.

(from the purdue math department's problem of the week series).

Here's a close cousin of your question - A, B, C, D are four distinct points in 3 space such that the sum of the angles ABC, BCD, CDA and DAB is 360 degrees. Does it then follow that all four points lie in the same plane? Prove this holds or give a counter-example.

by MaxEntropy_Man Thu Feb 16, 2012 7:12 pm

bw: will give it a shot. aside: this is a bit odd. the links to my solution don't work unless i am logged in. is this how it's supposed to be?

by Kris Thu Feb 16, 2012 11:42 pm

blabberwock wrote:
MaxEntropy_Man wrote:A,B,C,D are four distinct points in three space. Suppose each
of the angles ABC, BCD, CDA, and DAB are right angles.
Show that all four points lie in the same plane.

(from the purdue math department's problem of the week series).

Here's a close cousin of your question - A, B, C, D are four distinct points in 3 space such that the sum of the angles ABC, BCD, CDA and DAB is 360 degrees. Does it then follow that all four points lie in the same plane? Prove this holds or give a counter-example.

>>> When I read this yesterday, my first blush thoughts were as follows:

Assume first that A,B,C & D are on the XY plane. The sum of the angles is 360. Now let's rotate D up the z axis to a point D2, which results in a 3 dimensional quadrilateral ABCD2.

Angle B will not change and D2 will equal D (I think).

Angle A (or BAD2) and Angle C (or BCD2) will not equal BAD or BCD respectively. This being due to the angles BAD2 and BCD2 now having horizontal components and vertical components, the latter not having existed with BAD or BCD.

Therefore, the sum of the angles of the 3 dimensional quadrilateral cannot be 360. The delta between these angle sums (the new quadrilateral and the original collateral) will diminish as D2 gets closer to D and disappear only when D2 becomes D .

This leads to the quadrilateral with the sum of the angles being 360 being possible only when the 4 vertices are on the same plane.

I don't know if there are any flaws in the rationale above, but this led me to read up today on non-euclidean geometry and hyperbolic triangles etc. In the above scenario, the triangle ACD2 being a hyperbolic triangle will not add up to 180 degrees (quality of hyperbolic triangles, per my new found knowledge:)======> which means the new quad. will not add up to 360 degrees, which will lead to the necessity of ABCD having to be on the same plane.

P.S. My understanding of vectors/ non euclidean geometry is rudimentary at best, but this stuff kept me engrossed for a good couple of hours today. Thaks to you and Max for posting this stuff.

by Guest Fri Feb 17, 2012 1:32 am

Kris, your argument is quite convincing. Thanks.

by Guest Fri Feb 17, 2012 1:41 am

MaxEntropy_Man wrote: aside: this is a bit odd. the links to my solution don't work unless i am logged in. is this how it's supposed to be?

very odd indeed! your links don't work if i am logged out. many other links in this site don't work either in logged out mode. i suppose this is how it's supposed to be.

by doofus_maximus Sun Feb 19, 2012 3:04 am

you need to logged in to access the links.

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