Let 
be a tangential quadrilateral, 
meets 
at 
. Construct a circle (G) center at . 
are intersection of four polar line of to 
. Easily show that 
are parallelogram. 
meets 
at 
.
Prove that
, and center of circumcribed 
are collinear.
The polar of
meet at 
; The polar of 
meet at 
Prove that
are tangential quadrilateral similar (but not homothetic)
http://www.geogebratube.org/student/m79066
Solution by: Luis González
are then the poles of 
WRT 
and 
are the polars of 
and 
WRT 
is the pole of 
WRT 
But, as is the polar of WRT 
we have 
and therefore 
are collinear.
The second proposition is not completely true. is indeed tangential but not similar to in general.
Let
be the projections of on and 
the projections of on 
Then 
are the polars of 
WRT 
i.e. 
are the inverses of WRT 
Hence, from cyclic 
we get 
Similarly 
so are all homothetic with center 
Dilatate carrying 
onto 
i.e. 
(note here that ABCD and ABLI are not similar). If 
touches 
at 
and 
cuts 
at 
then 
and 
are homothetic is B- isosceles 
Similarly 
there is a circle 
touching 
at 
By Newton's theorem (degenerate Brianchon theorem) in 
the lines 
and 
concur, thus by the converse of Newton's theorem in 
it follows that 
touches 
i.e. 
is tangential the original is tangential.
be a tangential quadrilateral, meets at . Construct a circle (G) center at . are intersection of four polar line of to . Easily show that are parallelogram. meets at . Prove that
, and center of circumcribed are collinear. The polar of
meet at ; The polar of meet at Prove that
are tangential quadrilateral similar (but not homothetic)http://www.geogebratube.org/student/m79066
Solution by: Luis González
are then the poles of WRT and are the polars of and WRT is the pole of WRT But, as is the polar of WRT we have and therefore are collinear.The second proposition is not completely true.
is indeed tangential but not similar to in general.Let
be the projections of on and the projections of on Then are the polars of WRT i.e. are the inverses of WRT Hence, from cyclic we get Similarly so are all homothetic with center Dilatate
carrying onto i.e. (note here that ABCD and ABLI are not similar). If touches at and cuts at then and are homothetic is B- isosceles Similarly there is a circle touching at By Newton's theorem (degenerate Brianchon theorem) in the lines and concur, thus by the converse of Newton's theorem in it follows that touches i.e. is tangential the original is tangential.
Không có nhận xét nào:
Đăng nhận xét