103 Trigonometry Problems

Think Outside the Box
1. Trigonometric Fundamentals 33
A homothety (or central similarity, ordilation) is a transformation that fixes one
point O (called its center) and maps each point P to a point P ′ for which O,P, and
P ′ are collinear and the ratio OP : OP ′ = k is constant (k can be either positive or
negative). The constant k is called the magnitude of the homothety. The point P ′ is
called the image of P , and P the preimage of P ′ .
We can now answer our previous question. As shown in Figure 1.34, we first construct
a square BCE 2 D 2 outside of triangle ABC. (With compass and straightedge,
it is possible to construct a line perpendicular to a given line. How?) Let lines AD 2
and AE 2 meet segment BC at D and E, respectively. Then we claim that D and
E are two of the vertices of the square that we are looking for. Why? If line D 2 E 2
intersects lines AB and AC at B 2 and C 2 , then triangles ABC and AB 2 C 2 are homothetic
(with center A); that is, there is a dilation centered at A that takes triangle
ABC, point by point, to triangle AB 2 C 2 . It is not difficult to see that the magnitude
of the homothety is |AB 2|
|AB|
= |AC 2|
|AC|
= |B 2C 2 |
|BC| . Note that square BCE 2D 2 is inscribed
in triangle AB 2 C 2 . Hence D and E, the preimages of D 2 and E 2 , are the two desired
vertices of the inscribed square of triangle ABC.
A
S
T
B
D
E
C
B2
D2
E2
C2
Figure 1.34.
Menelaus’s Theorem
While Ceva’s theorem concerns the concurrency of lines, Menelaus’s theorem is
about the collinearity of points.