189. Pascal's theorem and some nice and easy applications.
by Virgil Nicula, Dec 11, 2010, 10:58 AM
Pascal's theorem. 
is a cyclic hexagon (convex or not) 
.
=====================================================================================
Proposed problem 1 (Gerry Leversha - England). For the triangle
with the orthocenter 
denote 
, 
, 
and
midpoints
, 
, 
of the sides ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, ![$[CA]$](//latex.artofproblemsolving.com/4/5/c/45c1acd47628de406680d04c09fe6314c3847acf.png)
, ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
respectively. Prove that the points 
, 
and 
are colinearly.
Proof. Apply the Pascal's theorem to the hexagon
which is inscribed in the Euler's circle of 
.
Proposed problem 2 (theory). Let be an hexagon which is inscribed in the circle 
and for which exists 
.
For
denote the intersections 
, 
and 
. Prove that the points 
are colinearly.
Proof. Apply the Pascal's theorem to the hexagons
.
Remark. Let be a cyclical hexagon. Then 
. See here my message.
Proposed problem 3 (own). Let be a triangle with the circumcircle . Denote the intersections 
and 
. For
define the intersections 
, 
and 
. Prove that the points 
are colinearly.
Proof. Apply Pascal's theorem to
.
Proposed problem 4 (own). Let bw a triangle with orthocenter and circumcircle 
. Denote 
.
Prove that :
.
Proposed problem 5 (own, very nice). Let
with circumcircle and incircle 
. Denote 
. Prove that 
.
is a cyclic hexagon (convex or not) 
.
. Apply in 
the Menelaus' theorem to the transversals :
because 
.=====================================================================================
Proposed problem 1 (Gerry Leversha - England). For the triangle
with the orthocenter 
denote 
, 
, 
andmidpoints
, 
, 
of the sides ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, ![$[CA]$](http://latex.artofproblemsolving.com/4/5/c/45c1acd47628de406680d04c09fe6314c3847acf.png)
, ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
respectively. Prove that the points 
, 
and 
are colinearly.Proof. Apply the Pascal's theorem to the hexagon
which is inscribed in the Euler's circle of 
.Proposed problem 2 (theory). Let
be an hexagon which is inscribed in the circle 
and for which exists 
.For
denote the intersections 
, 
and 
. Prove that the points 
are colinearly.Proof. Apply the Pascal's theorem to the hexagons
.Remark. Let
be a cyclical hexagon. Then 
. See here my message.Proposed problem 3 (own). Let
be a triangle with the circumcircle . Denote the intersections 
and 
. For
define the intersections 
, 
and 
. Prove that the points 
are colinearly. Proof. Apply Pascal's theorem to
.Proposed problem 4 (own). Let
bw a triangle with orthocenter and circumcircle 
. Denote 
.Prove that :
.
. Apply the upper second problem such : in the first case to the hexagon 
; in the second case to the hexagon 
and the point 
.Proposed problem 5 (own, very nice). Let
with circumcircle and incircle 
. Denote 
. Prove that 
. 
that 
. Apply the Pascal's theorem to 
, what is truly (the power of 
w.r.t. 
and 
obtain that 
.
, then from the Pascal's theorem applied to 
obtain that 
.This post has been edited 95 times. Last edited by Virgil Nicula, Nov 22, 2015, 6:04 PM
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