MathDB

Given is a flat plane

V

containing a rectangular coordinate system

xOy

. We consider quartets of numbers

(p,q,r,s)

;

p\le 0

q \le 0

r \le 0

s \le 0

. On every quartet we add a point

S

from

V

in a way that is in the accompanying figure is displayed. In this figure

OP = p

PQ = q

QR = r

RS = s

\angle OPQ = \angle PQR = \angle QRS = 135^o

. (a) What is the set of the points of

V

, which are added to these quartets ? (b) Which of these points has been added to only one quartet? How many quartets have the other points been added? (c) What is the set of points added to the quartets for which

p + q = 1

and

r = s = 0

? (d) What is the set of points added to the quartets for which

p + 1 =

and

r + s = 1

?
[asy] unitsize(0.6 cm);
pair O, P, Q, R, S;
O = (0,0); P = (2,0); Q = P + 2*dir(45); R = Q + (0,2.5); S = R + 3*dir(135);
draw((-1,0)--(7,0)); draw((0,-1)--(0,8)); draw(P--Q--R--S);
label("

O

", O, SW); label("

P

", P, dir(270)); label("

Q

", Q, E); label("

R

", R, E); label("

S

", S, N); label("

X

", (7,0), E); label("

Y

", (0,8), N); [/asy]

Problem Statement