This one has a simple, intuitive proof. Imagine a long, hollow square-prism. Something like a cardboard tube with a square cross-section. Imagine lines connecting the diagonally-opposite corners of the opposite ends. Due to the symmetry of the tube these four lines will meet at a single point half-way along it.
Now imagine looking through the tube from one end to the other. You see two squares: a large square at the near end, and a smaller square at the far end. By subtly changing the direction the tube is pointing, the far square may move off centre so that the construction looks like the problem diagram. This will not alter the fact that the four lines meet at a single point. QED!
Julian your approach is amazing!! In a way you are using perspective lines to prove the claim. If I understand correctly, it is like using an observation from real-life to prove a purely-mathematical claim.
Your idea is very original and creative, but I am not sure if it holds as a complete proof.
This one has a simple, intuitive proof. Imagine a long, hollow square-prism. Something like a cardboard tube with a square cross-section. Imagine lines connecting the diagonally-opposite corners of the opposite ends. Due to the symmetry of the tube these four lines will meet at a single point half-way along it.
ReplyDeleteNow imagine looking through the tube from one end to the other. You see two squares: a large square at the near end, and a smaller square at the far end. By subtly changing the direction the tube is pointing, the far square may move off centre so that the construction looks like the problem diagram. This will not alter the fact that the four lines meet at a single point. QED!
Julian your approach is amazing!!
ReplyDeleteIn a way you are using perspective lines to prove the claim. If I understand correctly, it is like using an observation from real-life to prove a purely-mathematical claim.
Your idea is very original and creative, but I am not sure if it holds as a complete proof.
Great work!