We have earlier the discussion about the proof
of Droz-Farny theorem.
Here is one proof of Droz-Farny 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
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
where v(ST) denotes vector ST.
Since M,N,P are the midpoints of segments
M1M2,N1N2,P1P2 we have
From (1) and (2) we get
Linear independency of vectors HP1 and HP2
leads us to two conditions
These conditions are
In these both conditions k would exist if it is
If we apply Menelaus theorem on
triangle M1P1B and line CA and on
triangle M2P2B and line CA we get
From (3) and (4) we have that the
theorem would be proved if it is true that
Let D be angle BP1H.Then we have
Theorem is proved.
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