1. - Sri Handayani –
(06022681318040)
Geometry
Area and Equality
&
The Pythagorean
Theorem Revisited

2. Area and Equality
Euclid’s principles called common notions. Euclid states are
particularly important for his theory of area, and they are as
following:
1. Things equal to the same thing are also equal to one another
2. If equal are added to equals, the wholes are equal
3. If equals are subtracted from equals, the remainders are equal
4. Things that coincide with one another are equal to one another
5. The whole is greater than the part

3. “Equals” in Euclid’s Proposition 15 of book I : Vertically
opposite angles are equal.
Vertically opposite angles are the angles α shown below
α α
β

4. The square of a sum
Proposition 4 of book II
It states a property of squares and rectangles that we express by
the algebraic formula
(a+b)2 = a2 + 2ab + b2
If a line is cut at random, the square on the whole is equal to the
squares on the segments and twice the rectangle contained by the
segments
a
a
b
b ab
aba2
b2

5. Example:
Construct plane that express
of algebraic formula below:
p(q+s) = pq + ps sq
p
Below the solution:
Area ABCD = length x width
= (q + s) x p
= p (q + s)
Area ABCD = area AEFD + area EBCD
= (pxq) + (pxs)
= pq + ps
So, p(q+s) = pq + ps
A
C
B
D F
E

6. The Pythagorean theorem revisited
In book VI, Proposition 31 of elements, Euclid prove a
generalization of the Pythagorean theorem
A B
C
a
b
c2c1
c
α
α β
β

7. A B
C
a
b
c2c1
c
α
α β
β
A
CB C
D
A
Da
b
c
a c1
hence
b2 = cc1
b
c
=
c1
b

8. A B
C
a
b
c2c1
c
α
α β
A
BB C
D
C
Da
b
c
a
c2
hence
a2 = cc2
a
c
=
c2
a
β

9. b2 = cc1 …… (1)
a2 = cc2 …… (2)
+
a2 + b2 = cc1 + cc2
a2 + b2 = c (c1 + c2 )
a2 + b2 = c (c)
a2 + b2 = c2
That is the
Pythagorean Theorem

10. Straightedge and compass construction of square
roots
h
l 1
The length h of this
perpendicular is

11. Reference
Stillwell, John. 2005. The Four Pillars of Geometry. USA: Springer

12. Thank
You