Consider a convex quadrilateral with vertices at and and on each side draw a square lying outside the given quadrilateral, as in the picture below. Let and be the centers of those squares:

Guest Sep 14, 2019

#1

#2**0 **

[asy]

size(220);

import TrigMacros;

rr_cartesian_axes(-3,8,-2,8,complexplane=true,usegrid = false);

pair A,B,C,D;

A = (2,1.5);

B= (4,1.8);

C = (5.3,3);

D = (3,5.3);

draw(A--B--C--D--A);

draw(A--(A + rotate(-90)*(B-A))--(B + rotate(90)*(A-B))--B);

draw(B--(B + rotate(-90)*(C-B))--(C + rotate(90)*(B-C))--C);

draw(C--(C + rotate(-90)*(D-C))--(D + rotate(90)*(C-D))--D);

draw(D--(D + rotate(-90)*(A-D))--(A + rotate(90)*(D-A))--A);

pair WW = (B + (A + rotate(-90)*(B-A)))/2;

dot(WW);

pair X = (C + (B + rotate(-90)*(C-B)))/2;

dot(X);

pair Y = (D + (C + rotate(-90)*(D-C)))/2;

dot(Y);

pair Z = (A + (D + rotate(-90)*(A-D)))/2;

dot(Z);

//draw(WW--Y,red);

//draw(X--Z,blue);

dot("$a$",A,SW);

dot("$b$",B,2*E);

dot("$c$",C,E);

dot("$d$",D,NNW);

dot("$p$",WW,E);

dot("$q$",X,S);

dot("$r$",Y,N);

dot("$s$",Z,S);

[/asy]

Guest Sep 14, 2019