Let P be a projective plane satisfying P5 and P6. Let ABCD be a complete quadrangle with diagonal points P,Q, and R. So, P=AB (int)CD, Q=AC(int)BD, and R=AD(int)BC. Let X,Y,Z be the points given by H(A,B;P,X), H(A,C;Q,Y), and H(B,C;R,Z). Use P5 to show that x,y,z are collinear.
I understand all the definitions being used in the proof. However, I cannot find the points X, Y, and Z. I know that the proof using P5 would utilize the fact that the two triangles in the picture laid out are perspective from a point and that they are from a line also and this would show that X, Y, and Z are collinear. The specific parts are just giving me trouble. Thanks in advance.
By the definition of harmonic quadruples $X=AB\cap QR$, $Y=AC\cap PR$, $Z=BC\cap PQ$. Since $ACB\triangle$ and $RPQ\triangle$ are perspective from point $D$, they are perspective from a line, and this line contains $X$, $Y$ and $Z$.