Advanced High-School Mathematics

4 CHAPTER 1 Advanced Euclidean Geometry

3. In the diagram to the right, 4 ABC
   is a right triangle, segments [AB]
   and [AF] are perpendicular and
   equal in length, and [EF] is per-
   pendicular to [CE]. Set a =
   BC, b = AB, c = AB, and de-
   duce President Garfield’s proof^1 of
   the Pythagorean theorem by com-
   puting the area of the trapezoid
   BCEF.

1.2.3 Similarity

In what follows, we’ll see that many—if not most—of our results shall
rely on the proportionality of sides insimilar triangles. A convenient
statement is as follows.

Similarity. Given the similar tri-

angles 4 ABC∼4A′BC′, we have

that

A′B

AB

=

BC′

BC

=

A′C′

AC

.

C

A

C'

B

A'

Conversely, if

A′B

AB

=

BC′

BC

=

A′C′

AC

,

then triangles 4 ABC∼4A′BC′are similar.

(^1) James Abram Garfield (1831–1881) published this proof in 1876 in theJournal of Education
(Volume 3 Issue 161) while a member of the House of Representatives. He was assasinated in 1881
by Charles Julius Guiteau. As an aside, notice that Garfield’s diagram also provides a simple proof
of the fact that perpendicular lines in the planes have slopes which are negative reciprocals.