Advanced book on Mathematics Olympiad

4.1 Geometry 207

In this section we grouped problems that require only the knowledge of the theory
of lines and circles. Recall that the general equation of a line (whether in a Cartesian or
affine coordinate system) isax+by+c=0. That of a circle (in a Cartesian coordinate
system) is(x−h)^2 +(y−k)^2 =r^2 , where(h, k)is the center andris the radius. Let us
see two examples, one in affine and one in Cartesian coordinates. But before we do that
let us recall that a complete quadrilateral is a quadrilateral in which the pairs of opposite
sides have been extended until they meet. For that reason, a complete quadrilateral has
six vertices and three diagonals.

Example.Prove that the midpoints of the three diagonals of a complete quadrilateral are
collinear.

Solution.As said, we will work in affine coordinates. Choose the coordinate axes to be
sides of the quadrilateral, as shown in Figure 25.

O

(0, )

(0, )

( ,0)a ( ,0)b

c

d

Figure 25

Five of the vertices have coordinates( 0 , 0 ),(a, 0 ),(b, 0 ),( 0 ,c), and( 0 ,d), while the
sixth is found as the intersection of the lines through(a, 0 )and( 0 ,d), respectively,( 0 ,c)
and(b, 0 ). For these two lines we know thex- andy-intercepts, so their equations are

1

a

x+

1

d

y=1 and

1

b

x+

1

c

y= 1.

The sixth vertex of the complete quadrilateral has therefore the coordinates
(
ab(c−d)
ac−bd

,

cd(a−b)

ac−bd

)

.

We find that the midpoints of the diagonals are

(a

2

,

c

2

)

,

(

b

2

,

d

2

)

,

(

ab(c−d)

2 (ac−bd)

,

cd(a−b)

2 (ac−bd)

)

.