Advanced High-School Mathematics

6 CHAPTER 1 Advanced Euclidean Geometry

4 ABCand 4 A′BC′are similar.

Exercises

1. Let 4 ABC and 4 A′B′C′ be given with ABĈ = A′̂B′C′ and
   A′B′
   AB

=

B′C′

BC

. Then 4 ABC∼4A′B′C′.
2. In the figure to the right,
AD=rAB, AE=sAC.
Show that
Area 4 ADE
Area 4 ABC

=rs.

D

B C

E

A

3. Let 4 ABC be a given triangle and letY, Z be the midpoints of
   [AC],[AB], respectively. Show that (XY) is parallel with (AB).
   (This simple result is sometimes called theMidpoint Theorem)


4. In 4 ABC, you are given that
   AY
   Y C

=

CX

XB

=

BX

ZA

=

1

x

,

where x is a positive real number.

Assuming that the area of 4 ABC

is 1, compute the area of 4 XY Zas

a function ofx.

Z

Y

X

C

B

A

5. LetABCDbe a quadrilateral and letEFGHbe the quadrilateral
   formed by connecting the midpoints of the sides ofABCD. Prove
   thatEFGHis a parallelogram.