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May 21st, 2018, 10:31 PM   #1
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Joined: Jan 2018
From: Belgrade

Posts: 55
Thanks: 2

Prove that points are coplanar

Quote:
 Let $\displaystyle ABCDA_{1}B_{1}C_{1}D_{1}$ be the cube with volume of $\displaystyle 1000cm^{3}$. Let $\displaystyle E$, $\displaystyle F$, $\displaystyle G$, $\displaystyle H$, $\displaystyle I$ and $\displaystyle J$ be the middle points of edges $\displaystyle BC$, $\displaystyle CD$, $\displaystyle DD_{1}$, $\displaystyle D_{1}A_{1}$, $\displaystyle A_{1}B_{1}$ and $\displaystyle B_{1}B$, respectively. Prove that points $\displaystyle E$, $\displaystyle F$, $\displaystyle G$, $\displaystyle H$, $\displaystyle I$ and $\displaystyle J$ belong to the same plane (are coplanar), and calculate surface area and volume of the pyramid $\displaystyle AEFGHIJ$.
I managed to calculate surface area and volume, but my question is about proof of coplanarity. How to prove that (without using vectors)? What idea is valid (in which direction to go):
- that angle between two planes is constant;
- that diagonals of the obtained regular hexagon are of the same length as are in 2D;
- or something else? May 22nd, 2018, 08:46 AM #2 Senior Member   Joined: Feb 2010 Posts: 711 Thanks: 147 If you establish a coordinate system with a corner at $\displaystyle (0,0,0)$ then you can get coordinates $\displaystyle E(10,5,0)$, $\displaystyle F(5,10,0)$, $\displaystyle G(0,10,5)$, $\displaystyle H(0,5,10)$, $\displaystyle I(5,0,10)$, $\displaystyle J(10,0,5)$. All of which satisfy the plane $\displaystyle x+y+z=15$. Thanks from greg1313 and lua Tags coplanar, points, prove Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Loren Geometry 6 April 8th, 2018 01:58 PM rohankhude New Users 0 April 14th, 2017 06:03 AM VengRouge Applied Math 2 June 21st, 2016 05:14 PM LasLasso Linear Algebra 1 February 11th, 2015 01:42 PM julien Applied Math 15 June 16th, 2008 11:21 PM

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