Three squares are placed side by side:
geometry/three.squares New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Three squares are placed side by side:
___________B___________C___________D
| ,| '| , |
| ,' | ' | , |
| ,' | ' , | |
| ,' '| , | |
| ,' ', | | |
|,', ' | | |
A -----------------------------------E
Prove the angle sum, BAE = CAE + DAE.
Yes, it's simple with trig functions; can you do it with just basic
geometry?
[ Editor: problem statement reconstructed from the postings; the
original reads "These are three squares." after the diagram. ]
-- Bill Taylor <wft@math.canterbury.ac.nz>
Show solution
Take a grid of six squares, made up of three squares across, two down.
Vertices labelled: top row A,B,C,E; middle row F,G,H,I; bottom row
J,K,L,M. We want to show that the angle CFM is 45. But CF=CM and FCM
is 90. So we are done.
-- John Scholes <jscholes@kalva.demon.co.uk>
___________B___________C___________D
| ,| '| , |
| ,' | ' |` , |
| ,' | ' , | ` |
| ,' '| , | ` |
| ,' ', | | ` |
|,', ' | | ` |
A -----------------------------------E
| ` | | ` |
| ` | | ` |
| | ` | ` |
| | ` | ` |
| | | ` ` |
| | | ` `|
-----------------------------------F
Angle DFC = angle BCA, so FCA is a right angle. But AC = CF, triangle
ACF is a right isoceles triangle, so angle CAF = 45 degrees.
Also angle DAE = angle FAE. In other words,
angle BAE = 45 degrees = angle CAF = CAE + FAE = CAE + DAE.
-- David A. Karr (karr@cs.cornell.edu)