Sec. 11.8 Isogonal conjugates 237(iii)
(−→
OZ
)=−→
OZ,
(iv)
(−−→
OZ 1 +
−−→
OZ 2 )=
−−→
OZ 1
+−−→
OZ 2
,(v)
(k−→
OZ)=k(−→
OZ),
(vi)
(−→
OZ
)=−→
OZ,
(vii)
(−→
OZ
)=−→
OZ
⊥
,(viii)
(−→
OZ
)=−−→
OZ
⊥
.Prove that axial symmmetry in the line through the origin, with angle of incli-
nationα,isgivenby−−→
OZ′=cos2α−→
OZ
+sin2α−→
OZ
.11.7 For non-collinear pointsZ 1 ,Z 2 ,Z 3 , suppose that
−−→
OZ 4 =−−→
OZ 2 +(
−−→
OZ 1 −
−−→
OZ 2 )⊥,
−−→
OZ 5 =
−−→
OZ 2 −(
−−→
OZ 3 −
−−→
OZ 2 )⊥.
Show that thenZ 1 Z 5 ⊥Z 3 Z 4 , and in fact−OZ−→
5 −−OZ−→
1 =
(−−→
OZ 4 −−OZ−→ 3
)⊥
.
11.8 Prove thatδF(Z 1 ,Z 2 ,Z 3 )+δF(Z 1 ,Z 4 ,Z 3 )=δF(Z 1 ,Z 5 ,Z 3 ),where
(Z 1 ,Z 2 )↑(Z 4 ,Z 5 ).11.9 If a pointZbe taken on the circumcircle of an equilateral triangle
[Z 1 ,Z 2 ,Z 3 ], prove that whenZis on the opposite side ofZ 2 Z 3 fromZ 1 ,|Z,Z 1 |=
|Z,Z 2 |+|Z,Z 3 |.11.10 For a triangle[Z 1 ,Z 2 ,Z 3 ],ifZ 4 is the foot of the perpendicular fromZ 1 toZ 2 Z 3 ,
show thatZ 4 is the mid-point of the orthocentre and the point where the line
Z 1 Z 4 meets the circumcircle again.
11.11 Deduce the Pasch property of a triangle from Menelaus’ theorem.