Ponder This Challenge - February 2004 - Right isosceles triangle with integer sides

Ponder This Challenge:

This month's puzzle is from Denis Boris.

Let ABC be a primitive right triangle with integer sides and integer area. ("Primitive" means that the sides are relatively prime.)

Let DEF be an isosceles triangle with integer sides.

Suppose ABC and DEF have the same perimeter, and the same area.

Part 1
Find the lengths of the sides of ABC and of DEF.

Part 2
Prove your solution is unique.
(We do not know the solution to Part 2.)

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  Part 1 Solution
  ABC has sides 135, 352, 377;
  DEF has sides 132, 366, 366.
  The common perimeter is 864, and the common area 23760.

  Part 2 Solution, provided by Dan Dima [PDF]

  ---

  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Part 1 & 2: (None)
• Dan Dima (2.12.2004 @07:07:05 AM EDT)
• Michael Stoll (2.27.2004 @04:25:23 PM EDT)
• Part 1: (None)
• Tomas G Rokicki (2.6.2004 @2:24:04 PM EDT)
• Brian Bitto (2.6.2004 @3:36:53 PM EDT)
• Georgios Papoutsis (2.6.2004 @3:50:16 PM EDT)
• Mikhail Sogrin (2.6.2004 @4:15:39 PM EDT)
• Michael Swart (2.6.2004 @4:22:57 PM EDT)
• Derek Kisman (2.6.2004 @5:42:29 PM EDT)
• Gyozo Nagy (2.6.2004 @6:02:27 PM EDT)
• Aleksey Pinkin (2.6.2004 @7:33:18 PM EDT)
• Cai Zheming (2.6.2004 @7:41:16 PM EDT)
• Bertram Felgenhauer (2.6.2004 @7:55:23 PM EDT)
• Daniel Wood (2.6.2004 @9:16:29 PM EDT)
• Michael McMullen (2.7.2004 @3:19:03 AM EDT)
• Piotr Zielinski (2.7.2004 @5:24:15 PM EDT)
• Joseph R Chadwick (2.7.2004 @10:54:29 PM EDT)
• Michael Brand (2.8.2004 @8:35:00 AM EDT)
• Guler Amos (2.8.2004 @5:16:00 PM EDT)
• Patrick J LoPresti (2.8.2004 @8:45:51 PM EDT)
• John G Fletcher (2.9.2004 @12:44:45 AM EDT)
• Subramanian C A (2.9.2004 @3:48:15 AM EDT)
• Don Dodson (2.9.2004 @3:37:53 PM EDT)
• Kennedy (2.10.2004 @1:00:25 AM EDT)
• Eugene Vasilchenko (2.10.2004 @1:19:57 AM EDT)
• Pavel Ustinov (2.10.2004 @4:25:54 AM EDT)
• Joseph DiVincentis (2.10.2004 @7:46:40 AM EDT)
• Klaus Hoffman (2.10.2004 @9:06:13 AM EDT)
• Vladimir Sedach (2.10.2004 @4:43:00 PM EDT)
• Steven Schveighoffer (2.10.2004 @5:32:49 PM EDT)
• Victor Chang (2.11.2004 @2:07:30 AM EDT)
• George Ross (2.11.2004 @10:03:17 AM EDT)
• Prakash Chandra (2.11.2004 @5:57:25 PM EDT)
• Joe Fendel (2.12.2004 @7:49:07 PM EDT)
• John Ritson (2.12.2004 @9:14:09 PM EDT)
• Nikolay Archak (2.13.2004 @2:05:51 AM EDT)
• Paul Cleary (2.13.2004 @8:17:22 AM EDT)
• Fred Bone (2.13.2004 @9:41:03 AM EDT)
• Ervan Darnell (2.13.2004 @10:14:13 PM EDT)
• Grzegorz Lukasik (2.14.2004 @7:36:37 AM EDT)
• Clive Tong (2.14.2004 @12:55:00 PM EDT)
• Bob Webster (2.14.2004 @5:46:50 PM EDT)
• Daniel Chong Jyh Tar (2.15.2004 @10:31:26 AM EDT)
• Pritibhushan Sinha (2.15.2004 @10:58:23 AM EDT)
• Renaud Bauvin (2.15.2004 @11:02:46 AM EDT)
• Sean Henderson (2.15.2004 @3:54:34 PM EDT)
• Lukasz Bolikowski (2.15.2004 @9:34:41 PM EDT)
• Lakshmi Giridharan (2.15.2004 @11:05:58 PM EDT)
• Patrick Coleman (2.16.2004 @3:48:41 AM EDT)
• Manish Kumar Singh (2.16.2004 @7:32:21 AM EDT)
• David Llorens (2.16.2004 @2:55:00 PM EDT)
• R Nandakumar (2.16.2004 @11:17:54 PM EDT)
• Chuck Grant (2.16.2004 @11:55:47 PM EDT)
• Sorin Ostafiev (2.17.2004 @2:45:24 AM EDT)
• Francis Golding (2.17.2004 @4:15:52 AM EDT)
• Christopher Howe (2.18.2004 @7:42:01 AM EDT)
• Raj Mohan N (2.19.2004 @4:24:57 AM EDT)
• David Friedman (2.19.2004 @10:32:39 AM EDT)
• Micha Anholt (2.19.2004 @4:34:41 PM EDT)
• Senthil Vel Gunasekaran (2.20.2004 @4:52:42 AM EDT)
• Boghirnea Radu (2.20.2004 @6:14:18 AM EDT)
• Mihaela Toarta (2.20.2004 @7:13:03 AM EDT)
• Bob Stewart (2.20.2004 @1:49:14 PM EDT)
• Vishal Goel (2.21.2004 @10:13:23 AM EDT)
• Jon Wharf (2.21.2004 @4:18:13 PM EDT)
• Tim Paulden (2.22.2004 @12:19:30 PM EDT)
• Rob Harley (2.22.2004 @4:54:45 PM EDT)
• Vladimir Baltic (2.23.2004 @12:54:37 PM EDT)
• Philippe Fondanaiche (2.23.2004 @1:05:09 PM EDT)
• Mark Stubbings (2.24.2004 @2:13:41 AM EDT)
• Jack Rouse (2.24.2004 @8:19:41 AM EDT)
• Greg Green (2.24.2004 @12:49:47 PM EDT)
• Alan O'Donnell (2.24.2004 @3:59:15 PM EDT)
• John Lister (2.24.2004 @10:31:26 PM EDT)
• Kuzub Valentin (2.25.2004 @1:34:43 AM EDT)
• Hongli Xu (2.25.2004 @5:06:49 AM EDT)
• Peter Huggins (2.25.2004 @12:11:17 PM EDT)
• IQSTAR (2.26.2004 @12:20:00 PM EDT)
• Patrick Hohmeyer (2.26.2004 @9:10:07 PM EDT)
• Divyesh Dixit (2.27.2004 @12:18:26 AM EDT)
• Kumar (2.27.2004 @6:11:00 AM EDT)
• Alex Wagner (2.27.2004 @3:33:58 PM EDT)
• Manoj K Thoniyil (2.27.2004 @4:01:35 PM EDT)
• James Chapman (2.27.2004 @7:27:38 PM EDT)
• Milana Milosevic (2.28.2004 @3:00:24 AM EDT)
• Nathaniel R Dean (2.28.2004 @5:10:55 PM EDT)
• Zhang Duo (2.28.2004 @11:50:22 PM EDT)
• Paul Botham (2.29.2004 @4:47:36 PM EDT)