206 4 Geometry and Trigonometry
584.On the sides of the triangleABCconstruct in the exterior the rectanglesABB 1 A 2 ,
BCC 1 B 2 ,CAA 1 C 2. Prove that the perpendicular bisectors ofA 1 A 2 ,B 1 B 2 , and
C 1 C 2 intersect at one point.
585.LetABCDbe a convex quadrilateral. The lines parallel toADandCDthrough
the orthocenterHof triangleABCintersectABandBC, respectively, atPand
Q. Prove that the perpendicular throughH to the linePQpasses through the
orthocenter of triangleACD.
586.Prove that if the four lines through the centroids of the four faces of a tetrahedron
perpendicular to those faces are concurrent, then the four altitudes of the tetrahedron
are also concurrent. Prove that the converse is also true.
587.LetABCbe a convex quadrilateral,M, N∈ABsuch thatAM=MN=NB,
andP,Q∈CDsuch thatCP=PQ=QD. LetObe the intersection ofAC
andBD. Prove that the trianglesMOPandNOQhave the same area.
588.LetABCbe a triangle, withDandEon the respective sidesACandAB.IfM
andNare the midpoints ofBDandCE, prove that the area of the quadrilateral
BCDEis four times the area of the triangleAMN.
4.1.2 The Coordinate Geometry of Lines and Circles................
Coordinate geometry was constructed by Descartes to translate Euclid’s geometry into the
language of algebra. In two dimensions one starts by fixing two intersecting coordinate
axes and a unit on each of them. If the axes are perpendicular and the units are equal, the
coordinates are called Cartesian (in the honor of Descartes); otherwise, they are called
affine. A general affine change of coordinates has the form
(
x′
y′
)
=
(
ab
cd)(
x
y)
+
(
e
f)
, with(
ab
cd)
invertible.If the change is between Cartesian systems of coordinates, a so-called Euclidean change
of coordinates, it is required additionally that the matrix
(
ab
cd
)
be orthogonal, meaning that its inverse is equal to the transpose.
Properties that can be formulated in the language of lines and ratios are invariant
under affine changes of coordinates. Such are the properties of two lines being parallel
or of a point to divide a segment in half. All geometric properties are invariant under
Euclidean changes of coordinates. Therefore, problems about distances, circles, and
angles should be modeled with Cartesian coordinates.