g.m. in a right angled triangle

it is not too daunting to prove (or "provide a logical explanation") why the length of the altitude of a right angled triangle is the geometric mean of the two segments either side of the foot of the altitude (where it splits the hypotenuse)

this result in the Russian geometry texts of Kiselev (born 1852) is introduced after proportionality
(translated by Alexander Givental into English)
pdf version of the first book, 'planimetry'
and forms his proof of the 'pythagoras' theorem (in chapter 6, theorem 188)

it is simply established using two similar triangles

or you could use the intersecting chord theorem (if you are fortunate enough to know it) - usually established from similar triangles

how was this used as a method to create a square equal in area to a rectangle?

you could use pythagoras to establish the result
but if you used similarity, how could this result be used to prove the pythagoras theorem?

some means

the arithmetic mean of two numbers is

and the geometric mean of two numbers is
 

find the arithmetical mean and geometrical mean for:

a = 2, b = 18
a = 3, b = 27
a = 1, b = 81
a = 4, b = 16
a = 2, b = 50
a = 5, b = 20

• when is the arithmetical mean closest to the geometric mean?
• how close can they be?
• which one is larger?
• does this happen for larger sets of numbers?
• for just two numbers: prove, by squaring, that the arithmetical mean is larger than or equal to the geometric mean

harmonic mean

if one piece of data is an extreme outlier it is recommended to use the harmonic mean to more appropriately represent an 'average' for the data set (but there are problems if one of the items in the data set is zero...)

try this for some simple data sets: compare the arithmetical mean with the harmonic mean where one of the numbers is large or small compared with the rest

for just two numbers, the arithmetical, geometric and harmonic means (along with the root mean square) can be represented by the following lengths:

establish that these lengths are the various means
















the harmonic mean of two lengths occurs in the crossed ladders problem - for the height at which two crossed ladders 'meet' (i.e. 'h' from 'A' and 'B')