## [EMHL] Droz-Farny theorem again

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• Dear friends, We have earlier the discussion about the proof of Droz-Farny theorem. Here is one proof of Droz-Farny theorem. Theorem Let two orthogonal lines
Message 1 of 1 , Jan 25, 2004
Dear friends,
We have earlier the discussion about the proof
of Droz-Farny theorem.

Here is one proof of Droz-Farny theorem.
Theorem
Let two orthogonal lines p1,p2 pass through
orthocenter H and intersect BC,CA,AB at points
M1,N1,P1 and M2,N2,P2 ,then the midpoints
M,N,P of segments M1M2,N1N2,P1P2 are
collinear.
Proof.
Let we have the points in following order
B,M1,H1,C,M2 and C,N2,N1,A and P1,A,H3,P2,B
where H1 and H3 are the feet of altitudes AH and CH.
The points M,N,P are collinear if and only if exist
k such that

v(HP)=k*v(HM)+(1-k)*v(HN) (1)

where v(ST) denotes vector ST.
Since M,N,P are the midpoints of segments
M1M2,N1N2,P1P2 we have

2v(HM)=v(HM1)+v(HM2),
2v(HN)=v(HN1)+v(HN2), (2)
2v(HP)=v(HP1)+v(HP2).

From (1) and (2) we get

v(HP1)+v(HP2)=k*[v(HM1)+v(HM2)]+
(1-k)*[v(HN1)+v(HN2)]=
k*[-(M1H/HP1)v(HP1)-(M2H/HP2)v(HP2)]+
(1-k)*[(HN1/HP1)v(HP1)-(N2H/HP2)v(HP2)].

Linear independency of vectors HP1 and HP2

1+k*(M1H/HP1)-(1-k)*(HN1/HP1)=0
1+k*(M2H/HP2)+(1-k)(N2H/HP2)=0

These conditions are

N1P1+k*M1N1=0
N2P2+k*M2N2=0

In these both conditions k would exist if it is
true that

M1N1/N1P1=M2N2/N2P2 (3)

If we apply Menelaus theorem on
triangle M1P1B and line CA and on
triangle M2P2B and line CA we get

M1N1/N1P1=(AB/P1A)*(M1C/BC) ,
M2N2/N2P2=(AB/AP2)*(CM2/BC). (4)

From (3) and (4) we have that the
theorem would be proved if it is true that

AP2/P1A=CM2/M1C (5)

Let D be angle BP1H.Then we have

AP2/P1H=(AH3+H3P2)/(P1H3-AH3)=
(tanB+tanD)/(cotanD-tanB)=tan(B+D)*tanD

CM2/M1C=(H1M2-H1C)/(M1H1+H1C)=
(tan(B+D)-tanB)/(cotan(B+D)+tanB)=tan(B+D)*tanD.

Theorem is proved.

Best regards