Triangles Inscribed in Rectangles

When a triangle is inscribed in a rectangle, the vertices of that triangle are on the sides or vertices of that rectangle.  Sometimes when additional conditions are added, some interesting properties occur.

For example, if the inscribed triangle is an equilateral triangle, and it shares a common vertex with the rectangle that it is inscribed in, then the sum of the areas of the two outer triangles (that share the same vertex) is equal to the area of the third outer triangle.  This theorem was described by a mathematician named Honsberger in the late 20^th century.

If the Inscribed Triangle is an Equilateral Triangle,

then Area X + Area of Y = Area of Z

The amazing thing about this theorem is that it holds true for any angle θ between the bottom of the equilateral triangle and the bottom of the rectangle.  As long as the equilateral triangle can be inscribed in the rectangle, the areas of the two outer triangles add up to the area of the third outer triangle.

To prove this, first find the value of each of the angles of the outer triangles in terms of θ, the angle between the bottom of the equilateral triangle and the bottom of the rectangle, using the fact that the angle sum of a triangle is 180°, and that the sum of the angles forming a line is 180°, and that the sum of the angles forming a right angle is 90°.  This means that ∠ADE = 30° – θ, ∠AED = 60° + θ, ∠ADE = 30° – θ, ∠BEF = 60° – θ, ∠BFE = 30° + θ, and ∠CFD = 90° – θ. 

Second, use A = ½bh =  ½(c sin θ)(c cos θ) = ½ c² sin θ cos θ = ¼ c²(2 sin θ cos θ) = ¼ c² sin 2θ for the area of each right triangle, where c is the hypotenuse and θ is one of the acute angles.  This means that the area of X = ¼ s² sin 2(30° – θ), the area of Y = ¼ s² sin 2θ, and the area of Z = ¼ s² sin 2(30° + θ).  Therefore:

01	X + Y	the sum of the areas
02	= ¼ s2 sin 2(30° – θ) + ¼ s2 sin 2θ	substitution
03	= ¼ s2 (sin 2(30° – θ) + sin 2θ)	factor out ¼ s2
04	= ¼ s2 (sin (60° – 2θ) + sin 2θ)	distribute 2
05	= ¼ s2 (sin 60° cos 2θ – cos 60° sin 2θ + sin 2θ)	sin (a – b) = sin a cos b – cos a sin b
06	= ¼ s2 (√3/2 cos 2θ – ½ sin 2θ + sin 2θ)	sin 60° = √3/2, cos 60° = ½
07	= ¼ s2 (√3/2 cos 2θ + ½ sin 2θ)	combine like terms
08	= ¼ s2 (sin 60° cos 2θ + cos 60° sin 2θ)	sin 60° = √3/2, cos 60° = ½
09	= ¼ s2 (sin (60° + 2θ))	sin (a + b) = sin a cos b + cos a sin b
10	= ¼ s2 sin 2(30° + θ)	factor out 2
11	= Z	substitution

Another two fascinating theorems concerning a triangle inscribed in a rectangle occur when two sides of the rectangle are cut in the same ratio.  If a triangle is inscribed in a rectangle so that it shares a common vertex and cuts two consecutive sides of the rectangle in the same ratio, then the sum of the areas of the two outer triangles (that also share the same vertex) is equal to area of half the rectangle it is inscribed in, and the product of the areas of the two outer triangles (that also share the same vertex) is equal to the product of the areas of the third outer triangle and half the rectangle it is inscribed in.

If E and F Cut the Sides of AB and BC in the Same Ratio,

then Area of X + Area Y = Area of Half the Rectangle

and Area of X · Area of Y = Area of Z · Area of Half the Rectangle

The amazing thing about this theorem is that it holds true no matter where the side is cut.  As long as the second side is cut in the same ratio, the sum of the areas of the two outer triangles is equal to the area of the half the rectangle, and the product of the areas of the two outer triangles is equal to the product of the areas of the third outer triangle and half the rectangle.

To prove this, label the sides of the rectangle b and h, and the ratio k, so that the cut segments are kb and b – kb and kh and h – kh.  Then using A = ½bh, the area of X = ½kbh, the area of Y = ½b(h – kh), the area of Z = ½kh(b – kb), and the area of half the rectangle is ½bh.  Therefore:

01	X + Y	the sum of the areas
02	= ½kbh + ½b(h – kh)	substitution
03	= ½kbh + ½bh – ½kbh	distribute ½b
04	= ½bh	combine like terms
05	= Area of Half the Rectangle	substitution

And also:

01	XY	the product of the areas
02	= ½kbh ∙ ½b(h – kh)	substitution
03	= ¼b2hk(h – kh)	commutative
04	= ¼b2h2k(1 – k)	factor out h
05	= ¼bh2k(b – kb)	distribute b
06	= ½kh(b – kb) ∙ ½bh	commutative
07	= Z ∙ Area of Half the Rectangle	substitution

There are several amazing theorems concerning triangles inscribed in rectangles and the outer triangles that are formed.  If the triangle is equilateral, then no matter the angle between the side of the triangle and the side of the rectangle, the area of two of the outer triangles add up to the area of the third triangle.  If the triangle cuts two consecutive sides of the rectangle in the same ratio, then no matter what the ratio the sides are cut at, the sum of the areas of two of the outer triangles is equal to the area of half the rectangle it is inscribed in, and the product of the areas of two of the outer triangles is equal to the product of the areas of the third outer triangle and half the rectangle it is inscribed in.  All of these theorems show just how wonderful and amazing geometry is.

Perfect Numbers

A perfect number is a number which is equal to the sum of all its divisors less than itself.  For example, 6 is a perfect number because all of its divisors less than itself are 1, 2, and 3; and 1 + 2 + 3 = 6.  The number 28 is another perfect number because all of its divisors less than itself are 1, 2, 4, 7, and 14; and 1 + 2 + 4 + 7 + 14 = 28.

Another way to define a perfect numbers is to state that is a number whose sum of all of its factors (including itself) is equal to twice its value.  For example, 6 is a perfect number because all of its factors are 1, 2, 3, and 6; and 1 + 2 + 3 + 6 = 12, which is twice its value.  The number 28 is another perfect number because all of its factors are 1, 2, 4, 7, 14, and 28; and 1 + 2 + 4 + 7 + 14 + 28 = 56, which is also twice its value.

A visual representation of the perfect number 6

(from Wikipedia)

A computer can be used to quickly find other perfect numbers.  The following program goes through every integer (less than a user-defined integer), finds its factors, adds them up, and displays the ones that are perfect along with its factors:

01	# Perfect Numbers
02	# Python 2.7.3
03	# To find all perfect numbers less than n, type "perfects(n)".  For example,
04	#   to find all perfect numbers less than 1000, type "perfects(1000)".
05
06	# prints a string with a hanging indentation
07	import textwrap
08	def hprint(string):
09	print textwrap.fill(textwrap.dedent(
10	string).strip(), initial_indent = "", subsequent_indent = "  ")
11
12	# returns a list of factors of n
13	def factor(n):
14	arrFactors = []
15	# go through all the numbers < or = the square root of n
16	for i in range(1, int(n ** 0.5) + 1):
17	# if the numbers divides into n, add it to the list
18	if (n % i == 0):
19	arrFactors.append(i)
20	# it is different, add the other factor to the list, too
21	if (i != n / i):
22	arrFactors.append(n / i)
23	# return the sorted list
24	return sorted(arrFactors)
25
26	# returns the sum of all the factors of n
27	def o(n):
28	intO = 0
29	# get the factors of n
30	arrFactors = factor(n)
31	# go through all the factors and add them up
32	for i in range(0, len(arrFactors)):
33	intO += arrFactors[i]
34	# return the sum
35	return intO
36
37	# determines if n is a perfect number
38	def perfect(n):
39	# if sigma of n (the sum of the factors) equals 2n, then it is a
40	#  perfect number, so display it and its factors, and return True
41	if (o(n) == 2 * n):
42	hprint(str(n) + ": " + str(factor(n)))
43	return True
44	# if not, return False
45	else:
46	return False
47
48	# find all the perfect numbers less than n
49	def perfects(n):
50	# go through all the numbers between 1 and n and test them for being
51	#   a perfect number
52	for i in range(1, n):
53	perfect(i)

Typing in “perfects(10000)” results in all the perfect numbers less than 10,000, and there are only 4: 6, 28, 496, and 8128.

>> perfects(10000)

6: [1, 2, 3, 6]

28: [1, 2, 4, 7, 14, 28]

496: [1, 2, 4, 8, 16, 31, 62, 124, 248, 496]

8128: [1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128]

Amazingly, the Ancient Greeks knew about all 4 of these perfect numbers in an age without computers.  In fact, around 300 B.C. Euclid proved that if 2^p – 1 is prime, then 2^{p – 1}(2^p – 1) is a perfect number.  For example, 3 is a prime number for p = 2 (2^p – 1 = 2² – 1 = 3), so 2^{p – 1}(2^p – 1) = 2^{2 – 1}(2² – 1) = 2(3) = 6 is a perfect number.  Also, 7 is a prime number for p = 3 (2^p – 1 = 2³ – 1 = 7), so 2^{p – 1}(2^p – 1) = 2^{3 – 1}(2³ – 1) = 4(7) = 28 is a perfect number.  Euclid’s proof can be found here (Proof PNP01).

Euclid

Since many perfect numbers are in the form 2^{p – 1}(2^p – 1), the following functions can be added to improve the above computer program to test 2^{p – 1}(2^p – 1) as a perfect number for any given p value:

55	# determines if (2 ** (p - 1)) * (2 ** p - 1) is a perfect number
56	def pperfect(p):
57	if perfect((2 ** (p - 1)) * (2 ** p - 1)) == False:
58	return False
59
60	# find all the perfect numbers in the form of (2 ** (p - 1)) * (2 ** p - 1)
61	#   less than the given value of p
62	def pperfects(p):
63	# go through all the numbers between 1 and p and test
64	#   (2 ** (p - 1)) * (2 ** p - 1) for being a perfect number
65	for i in range(1, p):
66	pperfect(i)

Typing in “pperfects(25)” quickly gives 3 more perfect numbers 33,550,336 and 8,589,869,056 and 137,438,691,328:

>> pperfects(25)

6: [1, 2, 3, 6]

28: [1, 2, 4, 7, 14, 28]

496: [1, 2, 4, 8, 16, 31, 62, 124, 248, 496]

8128: [1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128]

33550336: [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096,

  8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896,

  4193792, 8387584, 16775168, 33550336]

8589869056: [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096,

  8192, 16384, 32768, 65536, 131071L, 262142L, 524284L, 1048568L,

  2097136L, 4194272L, 8388544L, 16777088L, 33554176L, 67108352L,

  134216704L, 268433408L, 536866816L, 1073733632L, 2147467264L,

  4294934528L, 8589869056L]

137438691328: [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048,

  4096, 8192, 16384, 32768, 65536, 131072, 262144, 524287L, 1048574L,

  2097148L, 4194296L, 8388592L, 16777184L, 33554368L, 67108736L,

  134217472L, 268434944L, 536869888L, 1073739776L, 2147479552L,

  4294959104L, 8589918208L, 17179836416L, 34359672832L, 68719345664L,

  137438691328L]

Unfortunately, testing p ≥ 29 results in a “MemoryError” on my current personal computer, but supercomputers can be used to find more perfect numbers.

Euclid’s proof showed that perfect numbers are directly related to prime numbers in the form of 2^p – 1, and these prime numbers later became known as Mersenne primes, named after the French mathematician who studied them extensively in the 17^th century.  Over the years since, there have been many clever tricks developed to find larger and larger Mersenne prime numbers with the help of computers (see here for more details), and currently the largest known Mersenne prime is 2^74,207,281 – 1, a number with 22,338,618 digits!  According to Euclid’s proof, since 2^74,207,281 – 1 is a prime number, 2^74,207,280(2^74,207,281 – 1) must be a perfect number, currently making it the largest known perfect number, a number with 44,677,235 digits!

Mersenne

Interesting Facts

Here are some more interesting facts about perfect numbers:

-          Euler proved that all even perfect numbers must be in the form of 2^{p – 1}(2^p – 1) (Proof PNP02).

Euler

-          There are no known odd perfect numbers.  Computers have verified that there are no odd perfect numbers below 10¹⁵⁰⁰.

-          It is not known if there is an infinite number of perfect numbers.  (It is also not known if there is an infinite number of Mersenne primes.  Proving one would prove the other.)

-          All even perfect numbers (in the form of 2^{p – 1}(2^p – 1)) can be represented in binary as p ones followed by p − 1 zeros (Proof PNP03):

perf	binary
6	110
28	11100
496	111110000
8128	11111111000000

-          All even perfect numbers (in the form of 2^{p – 1}(2^p – 1))are the sum of all the integers from  1 to 2^p – 1 (also called 2^p – 1 triangular numbers) (Proof PNP04):

perf	Sum
6	1 + 2 + 3
28	1 + 2 + 3 + 4 + 5 + 6 + 7
496	1 + 2 + 3 + … + 29 + 30 + 31
8128	1 + 2 + 3 + … + 125 + 126 + 127

-          All even perfect numbers except for 6 (in the form of 2^{p – 1}(2^p – 1)) are the sum of the first 2^(p−1)/2 odd cubes (Proof PNP05):

perf	Sum
28	13 + 33
496	13 + 33 + 53 + 73
8128	13 + 33 + 53 + 73 + 93 + 113 + 133 + 153

-          All even perfect numbers except for 6 (in the form of 2^{p – 1}(2^p – 1)) are in the form of 1 + 9(T_(2^p-2)/3), where T_(2^p-2)/3 is the sum of all the integers from 1 to ^{2^p – 2} / ₃ (also called ^{2^p – 2} / ₃ triangular numbers) (Proof PNP06):

perf	sum
28	1 + 9(1 + 2)
496	1 + 9(1 + 2 + 3 + … + 8 + 9 + 10)
8128	1 + 9(1 + 2 + 3 + … + 40 + 41 + 42)

-          All even perfect numbers except 6 have a digital root of 1 (to find a digital root of a number, add the digits, then add the digits of the resulting number, then repeat the process until there is a single digit) (Proof PNP07):

perf	digit sum
28	2 + 8 = 10, 1 + 0 = 1
496	4 + 9 + 6 = 19, 1 + 9 = 10, 1 + 0 = 1
8128	8 + 1 + 2 + 8 = 19, 1 + 9 = 10, 1 + 0 = 1

-          The sum of reciprocals of all the divisors of every perfect number is equal to 2 (Proof PNP08)

perf	reciprocal sum
6	1/1 + 1/2 + 1/3 + 1/6 = 2
28	1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2
496	1/1 + 1/2 + 1/4 + … + 1/248 + 1/496 = 2
8128	1/1 + 1/2 + 1/4 + … + 1/4064 + 1/8128 = 2

                                                                                                                                            

-          Any number raised to a perfect number exponent is equal to the product of all its perfect roots.  For example, 6 is a perfect number, 2⁶ = 64, the perfect roots of 64 are ⁶√64 = 2, ³√64 = 4, and √64 = 8, and the product of 2, 4, and 8 is 2 ∙ 4 ∙ 8 = 64 (Proof PNP09).

perf	root product
6	6√x6 ∙ 3√x6 ∙ 2√x6 = x6
28	28√x28∙14√x28∙7√x28∙4√x28∙2√x28 = x28
496	496√x496∙248√x496∙…∙4√x496∙2√x496 = x496
8128	8128√x8128∙4064√x8128∙…∙4√x8128∙2√x8128 = x8128

-          All even perfect numbers end in 6 or 8 (Proof PNP10).  All even perfect numbers with more than 1 digit end in 16, 28, 36, 56, 76, or 96 (Proof PNP11).

perf	last digit	last 2 digits
6	6	n/a
28	8	28
496	6	96
8128	8	28

Related Concepts

Like perfect numbers, there are many other types of numbers defined by the sum of their divisors:

-          In the book Cutting for Stone by Abraham Verghese, one of the characters explains about a set of numbers whose sum of divisors is a perfect square:

“’Sixty-six is my second-favorite number,’ Shiva said. ‘Pray, why is it your second favorite?’ said Bailey. ‘Because if you take the numbers you can divide into sixty-six, including sixty-six, and add them up, what you have is a square.’ Mr. Bailey couldn't resist. He wrote down 1, 2, 3, 6, 11, 22, 33, and 66 – all the numbers that went into 66 – and then he totaled. What he got was 144, at which point both he and Shiva said, ‘Twelve squared!’ ‘That's what makes sixty-six special,’ Shiva said. ‘It's also true of three, twenty-two, sixty-six, seventy – their divisors add up to a square.’”

Cutting for Stone

-          An abundant number is a number in which the sum of all its divisors less than itself is greater than the number itself.  For example, 12 is an abundant number because its divisors less than itself are 1, 2, 3, 4, and 6; and 1 + 2 + 3 + 4 + 6 = 16 which is greater than 12.

-          A deficient number is a number in which the sum of all its divisors less than itself is less than the number itself.  For example, 8 is a deficient number because its divisors less than itself are 1, 2, and 4; and 1 + 2 + 4 = 7 which is less than 8.

-          Amicable numbers are a pair of numbers in which the sum of all the divisors less than itself of the first number is equal to the second number, and the sum of all the divisors less than itself of the second number is equal to the first number.  For example, 220 and 284 are amicable numbers because all of the factors less than itself of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110; and  1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284; and all the factors less than itself of 284 are 1, 2, 4, 71, and 142; and 1 + 2 + 4 + 71 + 142 = 220.  Other amicable numbers include 1184 and 1210, 2620 and 2924, 5020 and 5564, 6232 and 6368, and many more.

Conclusion

Perfect numbers have a simple definition but a wealth of related facts.  Yet despite the plethora of postulates concerning perfect numbers, there is still much to prove, including whether or not an odd perfect number exists, and whether or not there is an infinite number of perfect numbers.  Hopefully these mysteries will inspire mathematical works in the future.

Many of the facts about perfect numbers presented in this article were inspired by information found at Wolfram Mathematics and Wikipedia.