 Subject: Re: Formula for the height
Date: Sat, 29 Nov 1997 08:27:02 -0500
From: Alex Bogomolny

Dear Denis:

Your formula is good but the question is ill conceived. There is no standard formula for the height in a triangle. Mathematics would be boring if there were standard formulas. There is a great variety of them and it's up to you to choose the best suited for your purposes, or come up with a new one.

I can tell you this. Your formula is correct regardless of how you arrived at it. It also follows from the cosine law. This would be a direct verification. There is an indirect one:

Your formula is not symmetric with respect to b and c although in reality (at a glance) AD relates to b and c in the same manner and differently form its relation to a. Therefore, the formula obtained from yours by swapping b and c must also be correct. See that this is so

AD2 = sqrt(b2 - ((a2 + c2 - b2) / 2a)2).

Of the formulas I know, there is one most reminiscent of yours. It's the one that leads to (or is a consequence of) Heron's formula for the area of a triangle:

where p is the semiperimeter: p = (a+b+c)/2.

Glad you enjoy tinkering with math.

Sincerely,
Alexander Bogomolny 