Areas of Triangles – a View for Miles
I read that several solvers approached the Areas of Triangles puzzle by seeking out Heronian triples on the Internet and bypassing the intended logical deductions. I feel that this was a shame, but otherwise I have no problem with it. Perhaps this could have been forestalled by withholding Heron’s name in the preamble, but I felt that it would have been wrong to deny him the credit for the formula.
The formula itself is quite interesting, as initially it appears very far removed from the usual ‘half base times height’ rule, prompting the reaction ‘where on earth does it come from?’ The proof is readily available on the Internet, and it might prove an entertaining diversion for those so inclined, indeed it might surprise people how long it has been around.
I have long been interested in triangles (and surds too, but that is another story). Here are 3 nuggets: the area of the (3, 4, 5) triangle is 6, 12 is one height of the (13, 14, 15) triangle and all primitive Pythagorean triples contain a multiple of 3, a multiple of 4 and a multiple of 5. For the third one I remember once mentioning this to Jason Crampton as a conjecture, and he sent me a link on the Internet providing a proof. C’est la vie!
As for the intended solution, there were curious developments for solvers to spot, such as each primitive triple containing one even number and two odds, and each area having to be even, and these served as ‘rewards’ to quicken solving times. As for the extra step, this was meant to get solvers to consider the reverse procedure, obtaining the lengths of a triangle from its given area, which is quite a challenge (assuming of course no access to lists off the Internet!). As it happens I had chosen 924 because I had seen that it was associated with more than one triangle, however I was astonished when the editors revealed that there were ten of them!