Sec. 11.8 Isogonal conjugates 235
wherev=0asZ 1 ∈Z 2 Z 3. With this notation we have thatu+vı+λ 1 λ (^2) uu 2 −+vıv 2 is real
so thatv−λ 1 λ (^2) u (^2) +vv 2 =0, and henceλ 1 λ 2 =u^2 +v^2. We deduce thatλ 1 λ 2 =|Z^1 ,Z^2 |
2
|Z 3 ,Z 1 |^2.
Asλ 1 λ 2 >0, the pointsZ 4 andZ 5 must be both in[Z 2 ,Z 3 ]or both inZ 2 Z 3 [Z 2 ,Z 3 ].
11.8.2Concurrency..............................
Given non-collinear pointsZ 1 ,Z 2 ,Z 3 , suppose now that forZ 4 ,Z 5 ∈Z 2 Z 3 the half-
lines[Z 1 ,Z 4 ,[Z 1 ,Z 5 are isogonal conjugates with respect to the angle-support
|Z 2 Z 1 Z 3 ,forZ 6 ,Z 7 ∈Z 3 Z 1 the half-lines[Z 2 ,Z 6 ,[Z 2 ,Z 7 are isogonal conjugates with
respect to the angle-support|Z 3 Z 2 Z 1 ,andforZ 8 ,Z 9 ∈Z 1 Z 2 the half-lines[Z 3 ,Z 8 ,
[Z 3 ,Z 9 are isogonal conjugates with respect to the angle-support|Z 1 Z 3 Z 2 .Thenby
11.8.1, with an obvious notation we haveλ 4 λ 5 μ 6 μ 7 ν 8 ν 9 =|Z^1 ,Z^2 |
(^2) |Z 2 ,Z 3 | (^2) |Z 3 ,Z 1 | 2
|Z 3 ,Z 1 |^2 |Z 1 ,Z 2 |^2 |Z 2 ,Z 3 |^2 =1.
Now ifZ 1 Z 4 ,Z 2 Z 6 ,Z 3 Z 8 are concurrent, by Ceva’s theorem we haveλ 4 μ 6 ν 8 =1. It
follows thatλ 5 μ 7 ν 9 =1, which is the condition (11.5.1) forZ 1 Z 5 ,Z 2 Z 7 ,Z 3 Z 9 .Thenif
two of these are concurrent, the third must pass through their point of intersection.
11.8.3Symmedians..............................
If in 11.8.2[Z 1 ,Z 4 ,[Z 2 ,Z 6 ,[Z 3 ,Z 8 are the median half-lines of[Z 1 ,Z 2 ,Z 3 ],then
their isogonal conjugates[Z 1 ,Z 5 ,[Z 2 ,Z 7 ,[Z 3 ,Z 9 are called thesymmediansof the
triangle. Now the mid-pointsZ 4 ,Z 6 ,Z 8 lie in the segments[Z 2 ,Z 3 ],[Z 3 ,Z 1 ],[Z 1 ,Z 2 ],
respectively. Hence the pointsZ 5 ,Z 7 ,Z 9 lie in the segments[Z 2 ,Z 3 ],[Z 3 ,Z 1 ],[Z 1 ,Z 2 ],
respectively. By the cross-bar theorem the symmedian linesZ 1 Z 5 ,Z 2 Z 7 must meet and
thenZ 3 Z 9 must pass through their point of intersection. This point of concurrency is
called thesymmedian pointof the triangle.
To identify the symmedians fur-
ther, suppose thatObe the cen-
tre of the circumcircle of the
triangle[Z 1 ,Z 2 ,Z 3 ],andW 1 the
mid-point of {Z 2 ,Z 3 }.Letthe
line OW 1 meet the circumcir-
cle, on the opposite side ofZ 2 Z 3
fromZ 1 , at the pointU 1 ,[Z 1 ,W 1
meet the circle again atZ 4 ,and
the line through Z 4 parallel to
Z 2 Z 3 meet the circle again atZ 5.
Then|Z 2 Z 1 Z 3 and|Z 4 Z 1 Z 5 have
the same mid-lineZ 1 U 1 as each
other. Thus[Z 1 ,Z 5 is isogonal
conjugate of the median half-
line[Z 1 ,Z 4 and so is the corre-
sponding symmedian.
Z 1
Z 2
Z 3
O
U 1
Z 4
Z 5
W 1
Figure 11.22. Symmedians.