A Joint Venture To Neuberg Problem
by RSM, Jun 18, 2012, 5:27 AM
Here I will post solution to a problem which I and Potla were trying for more than a month. Last night we solved the problem.
So, let me state the problem at first:-
Given a triangle
and a point
, suppose,
are the reflections of
on
. Prove that
are concurrnt iff
is parallel to the Euler line of
where
is the isogonal conjugate of
wrt
.
The locus of such point
is the Neuberg Cubic of
. But in our proof we will not use any property of cubics. So let's call the locus of points
which satisfies the above concurrency fact as Neuberg locus(we are avoiding the term cubic).
At first we will prove 3 properties of the Neuberg locus. The properties are as follows:-
Property 1:-
Given a triangle
, if a point
lies on the Neuberg locus of
, then isogonal conjugate of
wrt
also lies on the Neuberg locus of
.
Proof:-
Suppose,
are the reflections of
on
. Suppose,
. Similarly define
. Under inversion wrt
with power
,
goes to the reflection of
on
and similar for others. Now note that, if
are the reflections of
on
, then
is homothetic to
. So if
are the reflections of
on
, then
are co-axial. But the center of
is the intersection point of
and
and similar for others. So
are collinear. So by Desargues' Theorem,
and
are perspective. So
lies on the Neuberg locus of
.
Property 2:-
Given a triangle
and a point
, prove that
lies on the Neuberg locus of
iff
lies on the Neuberg locus of the pedal triangle of
wrt
.
Proof:-
Suppose,
is the isogonal conjugate of
wrt
.
be the reflections of
on
. Note that,
is the circumcenter of
.
be the isogonal conjugate of
wrt
.
be the circumcenters of
. By some simple angle-chasing we get that
. So
is the isogonal conjugate of
wrt
and similar for others. But from property 1, if
lies on the Neuberg locus of
iff so does
. So
concurr. So
concurr. So
lies on the Neuberg locus of
. Now note that, if
is the pedal triangle of
wrt
and
is the isogonal conjugate of
wrt
, then the configuration
is homothetic to the configuration
. So
lies on the Neuberg locus of
. So again using property 1, we get that
lies on the Neuberg locus of
.
Corollary:-
Given a triangle
and a point
on its Neuberg Locus, suppose,
are the circumcenters of
. Prove that,
are concurrent. Call that concurrency point as
. If
is the isogonal conjugate of
wrt
, then define
for
similarly. Then
and
are isogonal conjugates wrt
.
Property 3:-
Given a triangle
and a point
, prove that
lies on the Neuberg locus(or circumcircle or line at infinity) of
iff the Euler lines of
are concurrent.
Proof:-
Let's ignore the case when
lies on circumcircle and line at infinity.
Let
be the centroids of the triangles
and
are the their circumcenters. Then we have
and
So, the homothety that maps
to
maps
to
Thus, ![$\frac{G_cK_1}{G_bK_1}=\frac{BK'_1}{CK'_1}.$](//latex.artofproblemsolving.com/4/9/b/49b5ccad9235b4dd2c6144fb70a74ea7d00e8c53.png)
Since the triangles
and
are perspective, so we get
which leads to
so that
and
are perspective. So using the corollary of Property 2 we get that
lies on the Neuberg cubic of
.
Before going to the proof of the main problem we will mention some lemmas that we will use in the proof.
Lemma 1:-
Given two triangles
and
(not homothetic), consider the set of all triangles(no two of them homothetic), so that both
and
are orthologic to them. Then the locus of the center of orthology of
and this set of triangles lie on a conic passing through
or the line at infinity.
Proof:-
On the sides of
take points
so that
is homothetic to
. Through
draw parallel to
to intersect
at
. Cyclically define
on
. Applying the converse of Pascal's theorem on the hexagon
we get that
lie on a conic(call this conic
).
. Clearly,
and
are homothetic. So
concurr. Since
is the harmonic conjugate of
wrt
, so if we draw parallels to
through
, then they intersect
at three collinear points. So there exists a conic passing through
so that the tangent at
is parallel to
and similar for
. Call this conic
. Take any point
on
. Through
draw a line
parallel to
and similarly define
. Note that
. So
lies on
and similar for others. So
concurr on
. Its easy to prove that its not true if
doesn't lie on
.
Lemma 2(Sondat's Theorem):-
Given two triangles
and
, such that they are orthologic and perspective, prove that the two centers of orthology are collinear with the center of perspectivity.
Proof:-
Suppose,
. Similarly define
such that
. Suppose,
concur at some point
.
Note that, if we define
wrt
using Lemma 1, then we get that, its the rectangular hyperbola passing through
and the orthocenter of
. Now note that,
and
are orthologic and
are the two centers of orthology. Note that,
passes through the orthocenter of
, but
wrt
is the rectangular hyperbola passing through
and the orthocenter of
. So
wrt
is same as
wrt
. But note that,
lies on
. So
lies on
wrt
. So
. So
are collinear.
Lemma 3:-
Given two points
and their isogonal conjugates wrt
be
, then suppose,
and
, then
is the isogonal conjugate of
wrt
.
Proof:-
So if the transformation in property 11 takes
to
, then it takes
to
. So
. So
lie on the same conic, which is the isogonal conjugate of
wrt
. So isogonal conjugate of
lies on
and similarly it lies on
. So
is the isogonal conjugate of
.
Also note that, if two points
are on
and
, such that
are isogonal conjugates, then
and
.
Some Observations:-
1. Given a triangle
and a point
, if the Euler lines of
are concurrent, then they concur on the Euler line of
.
Proof:-
Suppose,
are the circumcenters of
.
be the centroids of
. Suppose,
concur at some point
. Note that,
and
are perspective. Also note that, the perpendiculars from
to the sides of
concur at the circumcenter of
and the perpendiculars from
to the sides of
concur at the centroid of
. So by Sondat's Theorem
concur on the Euler line of
.
2. In the corollary of Property 2, we mentioned two points
. Using Sondat's theorem we get that,
lies on
and
lies on
. But since
and
are isogonal conjugates, so using Lemma 3, we get that
and
where
is the orthocenter of
.
Proof of The Neuberg Problem:-
Given a triangle
and a point
, suppose,
is the isogonal copnjugate of
wrt
.
be the circumcenters of
.
be the circumcenters of
. Suppose,
lies on the Neuberg locus of
. So
are concurrent. Clearly, it implies
are concurrent. Suppose,
is the circumcenter of
. Note that, the triangle formed by the circumcenters of
is homothetic to
. So if we draw parallels to the Euler lines of
through
they will meet at some point
. Similarly define
for
. Now note that,
is anti-parallel to
wrt
. So the Euler line of
is anti-parallel to the Euler line of
wrt
. Similar for others. So
is the isogonal conjugate of
wrt
. Note that, the triangle formed by the centroids of
is homothetic to
. So if we draw parallels to
through
, then they will concurr. So using lemma 1 we get that,
lies on
wrt
which is the rectangular hyperbola passing through
where
is the orthocenter of
. So
lies on
. Similarly,
lies on
. So using lemma 3, we get that,
and
. Now note that, if
is the concurreny point of
and
is the concurrency point of
, then from Observation 2, we have
and
. So
and
. From Observation 1, we get that the
is parallel to the Euler line of
. But now we have,
lies on
.
is the isogonal conjugate of
wrt
. So the line joining
and isogonal conjugate of
wrt
is parallel to the Euler line of
. So done.
The proof may look long, but the idea is very small. You can easily realize that after reading the proof.
Property 4:-
If
lies on the Neuberg locus of
, then
lies on the Neuberg Locus of
and similarly.
Proof:-
Note that, we have proved if the Euler lines of
are concurrent, then they concurr on the Euler line of
. So using property 3, we get that
lies on the Neuberg cubic of
.
Property 5:-
Given a triangle
and a point
, suppose,
are the intersection points of
with
.
are the orthocenters of
. Prove that
are concurrent iff
lies on the Neuberg cubic of
or the circles with diameter
or the line at infinity.
Proof:-
Suppose,
are the circumcnters of
. Note that,
are isogonal conjugates wrt
and similar for others. Suppose,
. Similarly, define
. Consider the case when
lies on the Neuberg cubic of
. Clearly,
concurr at some point
.
be the isogonal conjugate of
wrt
. Note that,
concurr at
. So applying converse of Brianchan's theorem on
we get that, there exists a conic touching
. So applying Brianchan's theorem on
we get that
are concurrent.
The other two cases are just special cases which can be verified easily.
Property 6(Concurrency of Brocard axes):-
Given a triangle
and point
, prove that Brocard axes of
are concurrent iff
lies on the Neuberg cubic of
or the circumcircle of
or the line at infinity.
Note that, the last two cases are trivial cases, so we will prove the first case only.
Proof:-
Before proving the problem we will prove two lemmas.
Lemma 1:-
Isotomic line of the Leomine axis of a triangle
is perpendicular to the Euler line of
.
Proof:-
Suppose,
is the Lemoine axis of
. Consider the circles with diameter
. Clearly, the orthocenter of
(call it
) lies on their radical axis. Also the circumcenter of
(call it
) lies on their radical axis. So they are co-axial with
as radical axis. But using Gaussian line theorem we get that the line joining the midpoints of
is parallel to the isotomic line of
wrt
. So the isotomic line of
wrt
is perpendicular to
.
Lemma 2:-
Let
be a triangle, and let
and
be two isotomic lines such that
and similars. Show that, if
is the point where the line through
parallel to
meets
then we have
.
Proof:-
Note that,
. And also
. But since
are isotomic points wrt
, so
and
. SO
.
Back to Main problem:-
Suppose,
are the circumcenter,orthocenter and symmedian point of
. Let
be the circumcenter of
. Note that, its enough to prove that,
. Suppose,
is the Lemoine axis of
. Then note that,
.
be the isotomic line of
. A line through
parallel to
intersects
at
. Using lemma 2 we get that,
. So
. But
. So
. But since the lines
are concurrent, so
. So we are done.
So, let me state the problem at first:-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ A',B',C' $](http://latex.artofproblemsolving.com/7/5/4/754219d697875ff40b8154492d0cc710b724675e.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ BC,CA,AB $](http://latex.artofproblemsolving.com/c/0/9/c09f0577d8be2f0c4f47a4ff32b47895bcf36b80.png)
![$ AA',BB',CC' $](http://latex.artofproblemsolving.com/a/7/4/a7450e1fe9fa12fcbb6bf94b246016faf2836db3.png)
![$ PP^* $](http://latex.artofproblemsolving.com/1/f/0/1f0af79133e540db4e47a895c03fc36a64f5f630.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
The locus of such point
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
At first we will prove 3 properties of the Neuberg locus. The properties are as follows:-
Property 1:-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Proof:-
Suppose,
![$ A',B',C' $](http://latex.artofproblemsolving.com/7/5/4/754219d697875ff40b8154492d0cc710b724675e.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ BC,CA,AB $](http://latex.artofproblemsolving.com/c/0/9/c09f0577d8be2f0c4f47a4ff32b47895bcf36b80.png)
![$ A_1=PA'\cap \odot A'B'C' $](http://latex.artofproblemsolving.com/b/4/2/b4233956bad8d1cb520a8f394923b261a8d79edb.png)
![$ B_1,C_1 $](http://latex.artofproblemsolving.com/2/2/d/22dc6fcf080e211628c9bc193f7a9ad0a5e2664a.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ PA'.PA_1 $](http://latex.artofproblemsolving.com/a/8/9/a892b51c3d25f03aa9b8dc7ef6a1df0a9b412e84.png)
![$ A $](http://latex.artofproblemsolving.com/a/3/8/a3838cb5b094421f0ff8b8f25affde76a00ddde6.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ B_1C_1 $](http://latex.artofproblemsolving.com/0/8/c/08c9bac564cebd75092ec217669410332c417ee4.png)
![$ A_2,B_2,C_2 $](http://latex.artofproblemsolving.com/2/b/2/2b2db335a4760bd777ab06fd5d8188a9b43b3855.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ BC,CA,AB $](http://latex.artofproblemsolving.com/c/0/9/c09f0577d8be2f0c4f47a4ff32b47895bcf36b80.png)
![$ \Delta A_2B_2C_2 $](http://latex.artofproblemsolving.com/4/f/0/4f039e574ebd2bbc6642c7a27e5f4e82256e259c.png)
![$ \Delta A_1B_1C_1 $](http://latex.artofproblemsolving.com/7/8/a/78a94c6ecb40e64c8ab68b79c4a4546253b1e671.png)
![$ A_3,B_3,C_3 $](http://latex.artofproblemsolving.com/6/1/3/6139a6afe44c799e540bec1a86d7ea9de6bb7512.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ B_2C_2, C_2A_2,A_2B_2 $](http://latex.artofproblemsolving.com/d/a/5/da5e4fdefc5da1758805f58ab033b2ec6bb1299e.png)
![$ \odot P^*A_2A_3, \odot P^*B_2B_3, \odot P^*C_2C_3 $](http://latex.artofproblemsolving.com/4/8/7/487827514f0231b519fa7b398ae6e50191ab17e1.png)
![$ \odot P^*A_2A_3 $](http://latex.artofproblemsolving.com/4/e/4/4e4c4ac679498ddacc770641f48f284ac78f197f.png)
![$ B_2C_2 $](http://latex.artofproblemsolving.com/0/1/2/012f2bc9941580a146c5973dddf1a08aec91ab13.png)
![$ BC $](http://latex.artofproblemsolving.com/d/1/b/d1b5a0ce1b62bd3c620eec70e59202f9f665b7e3.png)
![$ B_2C_2\cap BC,C_2A_2\cap CA,A_2B_2\cap AB $](http://latex.artofproblemsolving.com/2/5/3/253c0f3c20c3f5e435d9ba6bf36477526ed7e52e.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
![$ \Delta A_2B_2C_2 $](http://latex.artofproblemsolving.com/4/f/0/4f039e574ebd2bbc6642c7a27e5f4e82256e259c.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Property 2:-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Proof:-
Suppose,
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A',B',C' $](http://latex.artofproblemsolving.com/7/5/4/754219d697875ff40b8154492d0cc710b724675e.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ BC,CA,AB $](http://latex.artofproblemsolving.com/c/0/9/c09f0577d8be2f0c4f47a4ff32b47895bcf36b80.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ \Delta A'B'C' $](http://latex.artofproblemsolving.com/f/4/6/f4628dbfd40be9f189aa5a5a325fe770b076aaf0.png)
![$ P' $](http://latex.artofproblemsolving.com/a/2/c/a2c1a8c84d68d9f4fbb6c3cdceb425de1103470d.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ A'B'C' $](http://latex.artofproblemsolving.com/c/3/2/c32e9e3fe461f06e4c97b393863b44e056c7538f.png)
![$ A_1,B_1,C_1 $](http://latex.artofproblemsolving.com/b/e/7/be7834f6f1cec71c27289daae4e4804e84c21551.png)
![$ \Delta B'P'C', \Delta C'P'A', \Delta A'P'B' $](http://latex.artofproblemsolving.com/c/b/4/cb4be9aea7b51b8d55a00efb5bab7f3b33be0642.png)
![$ PA_1.PA=PA'^2 $](http://latex.artofproblemsolving.com/1/2/9/129b22460c46935d3886f8dee9f1d57084e9678b.png)
![$ A'A_1 $](http://latex.artofproblemsolving.com/2/9/c/29c8c0449e21bc681aaae9f07a39df56aa9226a9.png)
![$ A'A $](http://latex.artofproblemsolving.com/f/5/c/f5c4e8bcd3e943cfc289fcc2ae3138259f1a4a78.png)
![$ \angle B'A'C' $](http://latex.artofproblemsolving.com/9/f/f/9ff6ab79b5ce598716e08b06242e74b8ce8ac45e.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ A'A,B'B,C'C $](http://latex.artofproblemsolving.com/d/8/4/d841a49dffe48b22cfdb8b25b596721c1998bdab.png)
![$ A'A_1,B'B_1,C'C_1 $](http://latex.artofproblemsolving.com/1/5/8/158f851bd7b4cdf3c00989f6a222904131c3ded3.png)
![$ P' $](http://latex.artofproblemsolving.com/a/2/c/a2c1a8c84d68d9f4fbb6c3cdceb425de1103470d.png)
![$ \Delta A_1B_1C_1 $](http://latex.artofproblemsolving.com/7/8/a/78a94c6ecb40e64c8ab68b79c4a4546253b1e671.png)
![$ A_2B_2C_2 $](http://latex.artofproblemsolving.com/a/e/a/aea6c8a2eb4c062eef0bac189ca75a8b2a457c34.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ A_2B_2C_2 $](http://latex.artofproblemsolving.com/a/e/a/aea6c8a2eb4c062eef0bac189ca75a8b2a457c34.png)
![$ A_2B_2C_2Q $](http://latex.artofproblemsolving.com/4/d/0/4d05f427f7320b376d1fc5e74561115037c9abe3.png)
![$ A_1B_1C_1P' $](http://latex.artofproblemsolving.com/0/c/4/0c4e684112071d7b8b8a12176505c5381719bb41.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ \Delta A_2B_2C_2 $](http://latex.artofproblemsolving.com/4/f/0/4f039e574ebd2bbc6642c7a27e5f4e82256e259c.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ \Delta A_2B_2C_2 $](http://latex.artofproblemsolving.com/4/f/0/4f039e574ebd2bbc6642c7a27e5f4e82256e259c.png)
Corollary:-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P_a,P_b,P_c $](http://latex.artofproblemsolving.com/e/b/8/eb80a492ee9839d23c2593280a2b361444858372.png)
![$ \Delta BPC,\Delta CPA,\Delta APB $](http://latex.artofproblemsolving.com/4/7/8/47821c4f4328fbc5cfdb2f49c9281b97d4e7cc05.png)
![$ AP_a,BP_b,CP_c $](http://latex.artofproblemsolving.com/0/0/4/004a51c1884d252e97029c542886cfe5f99a2d70.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Property 3:-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ \Delta BPC,\Delta CPA,\Delta APB $](http://latex.artofproblemsolving.com/4/7/8/47821c4f4328fbc5cfdb2f49c9281b97d4e7cc05.png)
Proof:-
Let's ignore the case when
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
Let
![$G_a, G_b, G_c$](http://latex.artofproblemsolving.com/9/9/f/99fb11513b337a55d35e4b55571c161c738947ac.png)
![$BPC,CPA,APB.$](http://latex.artofproblemsolving.com/3/2/c/32cd9b86d1120b549beab34c7d96a152fef17bf2.png)
![$ O_a,O_b,O_c $](http://latex.artofproblemsolving.com/d/7/d/d7dc14fdc01649e2946902cc218c3c49265387a3.png)
![$G_bG_c\parallel BC$](http://latex.artofproblemsolving.com/7/6/8/768e3873ddc459243865705ff5145544c734b82f.png)
![$BG_c\cap CG_b=\text{midpoint}(AP).$](http://latex.artofproblemsolving.com/4/1/b/41b702c2c7d30c46153d1d16a00978638b21157b.png)
![$G_bG_c$](http://latex.artofproblemsolving.com/3/4/7/3472c4e51adea0356ac1eeb87d17b8c569bed943.png)
![$BC$](http://latex.artofproblemsolving.com/6/c/5/6c52a41dcbd739f1d026c5d4f181438b75b76976.png)
![$O_bO_c\cap G_bG_c\equiv K_1$](http://latex.artofproblemsolving.com/2/6/b/26b88601b9a1456ce69609a2d8d528c602272040.png)
![$O_bO_c\cap BC\equiv K_1'.$](http://latex.artofproblemsolving.com/2/b/9/2b94e5cd7bf4ab62f4879614687c9fd19ac58d15.png)
![$\frac{G_cK_1}{G_bK_1}=\frac{BK'_1}{CK'_1}.$](http://latex.artofproblemsolving.com/4/9/b/49b5ccad9235b4dd2c6144fb70a74ea7d00e8c53.png)
Since the triangles
![$G_aG_bG_c$](http://latex.artofproblemsolving.com/2/9/7/297cb6004671c4fecc84804ee209295df1e47197.png)
![$O_aO_bO_c$](http://latex.artofproblemsolving.com/b/a/f/bafd005d0f8f28f933f277749d85c03af5fb77de.png)
![$\prod_{cyc}\frac{G_cK_1}{G_bK_1}=-1,$](http://latex.artofproblemsolving.com/6/5/d/65da0ce7ff48fa75173a4ee4a46181925d8a9130.png)
![$\prod_{cyc}\frac{BK_1'}{CK_1'}=-1,$](http://latex.artofproblemsolving.com/d/8/e/d8ee1f628f50956338c83151fea57a272afec4d1.png)
![$ABC$](http://latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png)
![$O_aO_bO_c$](http://latex.artofproblemsolving.com/b/a/f/bafd005d0f8f28f933f277749d85c03af5fb77de.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
Before going to the proof of the main problem we will mention some lemmas that we will use in the proof.
Lemma 1:-
Given two triangles
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A'B'C' $](http://latex.artofproblemsolving.com/c/3/2/c32e9e3fe461f06e4c97b393863b44e056c7538f.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A'B'C' $](http://latex.artofproblemsolving.com/c/3/2/c32e9e3fe461f06e4c97b393863b44e056c7538f.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A,B,C $](http://latex.artofproblemsolving.com/e/a/0/ea0e0461ab301582db5a4dab58872f4c538e3750.png)
Proof:-
On the sides of
![$ BC,CA,AB $](http://latex.artofproblemsolving.com/c/0/9/c09f0577d8be2f0c4f47a4ff32b47895bcf36b80.png)
![$ C_1,A_1,B_1 $](http://latex.artofproblemsolving.com/a/8/b/a8b767b0922e6ab77a59f55e5c46d14523c92aa9.png)
![$ \Delta A_1B_1C_1 $](http://latex.artofproblemsolving.com/7/8/a/78a94c6ecb40e64c8ab68b79c4a4546253b1e671.png)
![$ \Delta A'B'C' $](http://latex.artofproblemsolving.com/f/4/6/f4628dbfd40be9f189aa5a5a325fe770b076aaf0.png)
![$ C_1 $](http://latex.artofproblemsolving.com/c/e/7/ce7983f47008c97f32cb152ee951c279e080fa61.png)
![$ AC $](http://latex.artofproblemsolving.com/d/2/f/d2f14bcb06d1fcb412e1996eae83c31540a32f6a.png)
![$ AB $](http://latex.artofproblemsolving.com/9/5/9/95956c313818c1b13a7bd08e6136df2b10b0ab0b.png)
![$ A_2 $](http://latex.artofproblemsolving.com/b/2/b/b2b73513eb199820fb5cb04c592f95e5a8e4e78f.png)
![$ B_2,C_2 $](http://latex.artofproblemsolving.com/5/6/f/56fc02ab26b0bc4f4c0f7549954e9155f9f3bfd0.png)
![$ BC,AC $](http://latex.artofproblemsolving.com/a/a/a/aaae29feb216d24628a455dd4768dadf81c045f8.png)
![$ A_1C_2B_1A_2C_1B_2 $](http://latex.artofproblemsolving.com/f/e/3/fe32ac186bdf695a123ccb5eb9c79190b27a6728.png)
![$ A_1,B_1,C_1,A_2,B_2,C_2 $](http://latex.artofproblemsolving.com/b/1/1/b11017e5ff2016f1c87d62ddbd1fcf64149854ec.png)
![$ \Gamma(A_1B_1C_1) $](http://latex.artofproblemsolving.com/8/4/e/84e3ae525b66bb7a5b658a6a86c57e62f53af2af.png)
![$ A_3=A_2C_1\cap B_2A_1, B_3=B_2A_1\cap B_1C_2, C_3=C_2B_1\cap A_2C_1 $](http://latex.artofproblemsolving.com/4/6/0/4601686cbdacd8d1c194bab8f6935b334e34d30a.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A_3,B_3,C_3 $](http://latex.artofproblemsolving.com/6/1/3/6139a6afe44c799e540bec1a86d7ea9de6bb7512.png)
![$ AA_3,BB_3,CC_3 $](http://latex.artofproblemsolving.com/5/f/c/5fc51c8b49668e335654c14f197a3fcafdf4d107.png)
![$ A_2A_1 $](http://latex.artofproblemsolving.com/3/7/d/37d75f48b8e26092b19a1460141fc67ae3ecdaf5.png)
![$ AA_3 $](http://latex.artofproblemsolving.com/0/6/2/062a46af3c3cf4c587a74254a0f6f01feac19f69.png)
![$ AB,AC $](http://latex.artofproblemsolving.com/5/3/b/53b164eda11acd6abe806e619f76d3cbca2449db.png)
![$ A_1A_2,B_1B_2,C_1C_2 $](http://latex.artofproblemsolving.com/e/0/e/e0ee383ede3451b764c76671f6b58d8b1cfe8027.png)
![$ A,B,C $](http://latex.artofproblemsolving.com/e/a/0/ea0e0461ab301582db5a4dab58872f4c538e3750.png)
![$ BC,CA,AB $](http://latex.artofproblemsolving.com/c/0/9/c09f0577d8be2f0c4f47a4ff32b47895bcf36b80.png)
![$ A,B,C $](http://latex.artofproblemsolving.com/e/a/0/ea0e0461ab301582db5a4dab58872f4c538e3750.png)
![$ A $](http://latex.artofproblemsolving.com/a/3/8/a3838cb5b094421f0ff8b8f25affde76a00ddde6.png)
![$ A_1A_2 $](http://latex.artofproblemsolving.com/b/5/a/b5a3a8f73a52f1a06962174db0c98d8d3095998f.png)
![$ B,C $](http://latex.artofproblemsolving.com/f/9/1/f91cea12f9cc92f60325007797c6219b981c82e7.png)
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
![$ X $](http://latex.artofproblemsolving.com/d/d/9/dd910b068ee563c04018031d949a7c56a571b0e3.png)
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
![$ A_1 $](http://latex.artofproblemsolving.com/e/9/9/e996532b539515f650041918047d508ca5a1feb8.png)
![$ l_a $](http://latex.artofproblemsolving.com/4/a/3/4a351c1952a9f2a7b3c174ae0d167203161af5e8.png)
![$ AX $](http://latex.artofproblemsolving.com/3/a/f/3afefdad2b8f23accf6c15834c41a7d26fb1dac2.png)
![$ l_b,l_c $](http://latex.artofproblemsolving.com/2/2/2/2228d42d8ef86ff4a631e5aed9ad0cac690d7faf.png)
![$ (l_a,A_1A_2;A_1C_2,A_1B_2)=(AX,A_1A_2;AC,AB) $](http://latex.artofproblemsolving.com/4/5/4/454dfde1265dd5e4a982c5569ff8e1ce164175aa.png)
![$ =(BX,BA;BC,B_1B_2)=(l_b,B_1A_2;B_1C_2,B_1B_2) $](http://latex.artofproblemsolving.com/a/4/b/a4b1efc90156979c93a8a5b4d7e8faa785ea6d34.png)
![$ l_a\cap l_b $](http://latex.artofproblemsolving.com/7/b/c/7bcc8f61869d4d90173385850752d3fd4e6404a4.png)
![$ \Gamma(A_1B_1C_1) $](http://latex.artofproblemsolving.com/8/4/e/84e3ae525b66bb7a5b658a6a86c57e62f53af2af.png)
![$ l_a,l_b,l_c $](http://latex.artofproblemsolving.com/0/5/5/05574dc994eb4669961b1c90b01a6d396fce01cb.png)
![$ \Gamma(A_1B_1C_1) $](http://latex.artofproblemsolving.com/8/4/e/84e3ae525b66bb7a5b658a6a86c57e62f53af2af.png)
![$ X $](http://latex.artofproblemsolving.com/d/d/9/dd910b068ee563c04018031d949a7c56a571b0e3.png)
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
Lemma 2(Sondat's Theorem):-
Given two triangles
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A'B'C' $](http://latex.artofproblemsolving.com/c/3/2/c32e9e3fe461f06e4c97b393863b44e056c7538f.png)
Proof:-
Suppose,
![$ AP\perp B'C', BP\perp C'A', CP\perp A'B' $](http://latex.artofproblemsolving.com/c/a/d/cadded6a7d3e09f8a845f8695f9494b2a3d0ca47.png)
![$ P' $](http://latex.artofproblemsolving.com/a/2/c/a2c1a8c84d68d9f4fbb6c3cdceb425de1103470d.png)
![$ A'P'\perp BC, B'P'\perp CA, C'P'\perp AB $](http://latex.artofproblemsolving.com/1/2/c/12c92194dfff5f4275ab4fda6d8433bfbe12c2d0.png)
![$ AA',BB',CC' $](http://latex.artofproblemsolving.com/a/7/4/a7450e1fe9fa12fcbb6bf94b246016faf2836db3.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
Note that, if we define
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
![$ \Delta A'B'C' $](http://latex.artofproblemsolving.com/f/4/6/f4628dbfd40be9f189aa5a5a325fe770b076aaf0.png)
![$ A,B,C,P $](http://latex.artofproblemsolving.com/b/6/f/b6f9d33b4c14dc032bcbd62e81cbe1f78f34c0ca.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ BPC $](http://latex.artofproblemsolving.com/e/f/4/ef438bddf61934d8314278b0cb669e3d178750d5.png)
![$ B'P'C' $](http://latex.artofproblemsolving.com/7/8/e/78e4ce912b613cc8c50c62828f90991e4d633128.png)
![$ A,A' $](http://latex.artofproblemsolving.com/8/f/7/8f7fbb9256210aef24fa50906a57228b6fb64cc3.png)
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
![$ \Delta BPC $](http://latex.artofproblemsolving.com/d/5/8/d584227099c306b0c6e8dd760d018d00b129150a.png)
![$ \Gamma(BPC) $](http://latex.artofproblemsolving.com/7/d/7/7d707b8e1bb975833914118b4af22bdc4d0d6c49.png)
![$ \Delta B'P'C' $](http://latex.artofproblemsolving.com/5/d/b/5dbbe9bcd12407d5f91f2f1d6b364b2d2758dc99.png)
![$ B,C,P,A $](http://latex.artofproblemsolving.com/c/4/5/c452c074822ef4f67d973373b2b962688393b822.png)
![$ \Delta BPC $](http://latex.artofproblemsolving.com/d/5/8/d584227099c306b0c6e8dd760d018d00b129150a.png)
![$ \Gamma(BPC) $](http://latex.artofproblemsolving.com/7/d/7/7d707b8e1bb975833914118b4af22bdc4d0d6c49.png)
![$ \Delta B'P'C' $](http://latex.artofproblemsolving.com/5/d/b/5dbbe9bcd12407d5f91f2f1d6b364b2d2758dc99.png)
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
![$ \Delta A'B'C' $](http://latex.artofproblemsolving.com/f/4/6/f4628dbfd40be9f189aa5a5a325fe770b076aaf0.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ \Gamma(BPC) $](http://latex.artofproblemsolving.com/7/d/7/7d707b8e1bb975833914118b4af22bdc4d0d6c49.png)
![$ B'P'C' $](http://latex.artofproblemsolving.com/7/8/e/78e4ce912b613cc8c50c62828f90991e4d633128.png)
![$ PQ\parallel P'Q $](http://latex.artofproblemsolving.com/d/a/2/da28d54c556d381a3d09d59adcda9246ed7de25e.png)
![$ P,P',Q $](http://latex.artofproblemsolving.com/4/3/e/43e7507579b27fa2d5cd9fcae2b2ee2d221b58ea.png)
Lemma 3:-
Given two points
![$ P,Q $](http://latex.artofproblemsolving.com/1/8/c/18cd0402e199ea3ecdbf0b4f0fb6ef6e5d926769.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P',Q' $](http://latex.artofproblemsolving.com/c/e/6/ce6e9230b9863eb88f5e2fd57f5e530938f5160f.png)
![$ X=PQ'\cap QP' $](http://latex.artofproblemsolving.com/e/c/6/ec6c2c7cfc4fb18a13ca266ff0b19a2733ea9f21.png)
![$ Y=PQ\cap P'Q' $](http://latex.artofproblemsolving.com/8/d/0/8d0b38eb44de4a568a220b605765a52f81eb9090.png)
![$ Y $](http://latex.artofproblemsolving.com/8/f/0/8f03b971560f85686771754e7a291a7def310954.png)
![$ X $](http://latex.artofproblemsolving.com/d/d/9/dd910b068ee563c04018031d949a7c56a571b0e3.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Proof:-
So if the transformation in property 11 takes
![$ P' $](http://latex.artofproblemsolving.com/a/2/c/a2c1a8c84d68d9f4fbb6c3cdceb425de1103470d.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ Q' $](http://latex.artofproblemsolving.com/f/f/8/ff89702d63b97ff40add18eb9f75fd5c3079a2ef.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ (P'A,P'B;P'C,P'Q')=(QA,QB;QC,QP) $](http://latex.artofproblemsolving.com/2/5/6/256776d32b92be2b7336c6732eb63d6bc823f3cd.png)
![$ \implies (P'A,P'B;P'C,P'Y)=(QA,QB;QC,QY) $](http://latex.artofproblemsolving.com/8/1/b/81bfeaa385dab6f5be3a6b982368d09df6ffdc60.png)
![$ A,B,C,Q,Y,P' $](http://latex.artofproblemsolving.com/9/4/b/94bc9558b366e2cf0b6b16289223ae3733a91433.png)
![$ PQ' $](http://latex.artofproblemsolving.com/3/5/6/356fa77fff80ae5c8aa40c35235dd138d6dffc14.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ Y $](http://latex.artofproblemsolving.com/8/f/0/8f03b971560f85686771754e7a291a7def310954.png)
![$ PQ' $](http://latex.artofproblemsolving.com/3/5/6/356fa77fff80ae5c8aa40c35235dd138d6dffc14.png)
![$ QP' $](http://latex.artofproblemsolving.com/6/2/3/62390ddb3186f3380c57796bdfd5e7d1752f7eea.png)
![$ X $](http://latex.artofproblemsolving.com/d/d/9/dd910b068ee563c04018031d949a7c56a571b0e3.png)
![$ Y $](http://latex.artofproblemsolving.com/8/f/0/8f03b971560f85686771754e7a291a7def310954.png)
Also note that, if two points
![$ X,Y $](http://latex.artofproblemsolving.com/d/a/5/da5adf5089bbdb26c7b8ad89830c005f2945f25e.png)
![$ PQ' $](http://latex.artofproblemsolving.com/3/5/6/356fa77fff80ae5c8aa40c35235dd138d6dffc14.png)
![$ PQ $](http://latex.artofproblemsolving.com/5/9/6/59667f482dbe426f5c691c03076590d756f53cf9.png)
![$ X,Y $](http://latex.artofproblemsolving.com/d/a/5/da5adf5089bbdb26c7b8ad89830c005f2945f25e.png)
![$ X=PQ'\cap QP' $](http://latex.artofproblemsolving.com/e/c/6/ec6c2c7cfc4fb18a13ca266ff0b19a2733ea9f21.png)
![$ Y=PQ\cap P'Q' $](http://latex.artofproblemsolving.com/8/d/0/8d0b38eb44de4a568a220b605765a52f81eb9090.png)
Some Observations:-
1. Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ BPC,CPA,APB $](http://latex.artofproblemsolving.com/5/a/2/5a232e185d2be3646029916ec00124a1771bf326.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Proof:-
Suppose,
![$ O_a,O_b,O_c $](http://latex.artofproblemsolving.com/d/7/d/d7dc14fdc01649e2946902cc218c3c49265387a3.png)
![$ \Delta BPC, \Delta CPA,\Delta APB $](http://latex.artofproblemsolving.com/2/6/5/26531c2ef703bb6ade2cc8fe5b68cb9d1f6c073c.png)
![$ G_a,G_b,G_c $](http://latex.artofproblemsolving.com/4/e/9/4e9cd6dfc202f8763d201487d6ba94c3a8075bc2.png)
![$ \Delta BPC,\Delta CPA,\Delta APB $](http://latex.artofproblemsolving.com/4/7/8/47821c4f4328fbc5cfdb2f49c9281b97d4e7cc05.png)
![$ O_aG_a,O_bG_b,O_cG_c $](http://latex.artofproblemsolving.com/4/3/e/43e08a084c6a71be4d7f931cbddfc4b373c1b5de.png)
![$ X $](http://latex.artofproblemsolving.com/d/d/9/dd910b068ee563c04018031d949a7c56a571b0e3.png)
![$ \Delta O_aO_bO_c $](http://latex.artofproblemsolving.com/b/4/e/b4e75b60b6a07ffbbc9e9e310d1952df36d0feef.png)
![$ \Delta G_aG_bG_c $](http://latex.artofproblemsolving.com/9/8/1/981eb2184dcb44d27c22e6c925a321b0dd0ffcb3.png)
![$ O_a,O_b,O_c $](http://latex.artofproblemsolving.com/d/7/d/d7dc14fdc01649e2946902cc218c3c49265387a3.png)
![$ \Delta G_aG_bG_c $](http://latex.artofproblemsolving.com/9/8/1/981eb2184dcb44d27c22e6c925a321b0dd0ffcb3.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ G_a,G_b,G_c $](http://latex.artofproblemsolving.com/4/e/9/4e9cd6dfc202f8763d201487d6ba94c3a8075bc2.png)
![$ O_aO_bO_c $](http://latex.artofproblemsolving.com/8/7/d/87dd30c4d0f664c0dcad81a8bd310c4fd3b3bdb4.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
![$ O_aG_a,O_bG_b,O_cG_c $](http://latex.artofproblemsolving.com/4/3/e/43e08a084c6a71be4d7f931cbddfc4b373c1b5de.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
2. In the corollary of Property 2, we mentioned two points
![$ Q,Q^* $](http://latex.artofproblemsolving.com/1/c/6/1c6c8dd6a8b5b4642538f7341f3897a1f044779a.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ OP $](http://latex.artofproblemsolving.com/7/b/d/7bda632508deee45dc4c44e740a68eca66e5179a.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ OP^* $](http://latex.artofproblemsolving.com/c/6/c/c6c53ee012bb891d66edd208215ec9f9c15a4db5.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ Q=OP\cap HP^* $](http://latex.artofproblemsolving.com/2/2/4/2240533cb0aff8a40814e8ea67cdf7e61057c1c0.png)
![$ Q^*=OP^*\cap HP $](http://latex.artofproblemsolving.com/d/1/2/d12c0102069648235429c0338241d2f6ec299fde.png)
![$ H $](http://latex.artofproblemsolving.com/b/2/1/b21b6cf85d4bdfe12f052f6d1451af9150b5ed8d.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Proof of The Neuberg Problem:-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P_a,P_b,P_c $](http://latex.artofproblemsolving.com/e/b/8/eb80a492ee9839d23c2593280a2b361444858372.png)
![$ \Delta BPC,\Delta CPA,\Delta APB $](http://latex.artofproblemsolving.com/4/7/8/47821c4f4328fbc5cfdb2f49c9281b97d4e7cc05.png)
![$ P^*_a,P^*_b,P^*_c $](http://latex.artofproblemsolving.com/a/4/4/a44ab30cc078f6cfae441d9a7fd917b2fa8eab52.png)
![$ \Delta BP^*C,\Delta CP^*A,\Delta AP^*B $](http://latex.artofproblemsolving.com/9/2/7/9273d9009a724f36153bdc5514658dbeb94f1a12.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ \Delta P_aP_bP_c $](http://latex.artofproblemsolving.com/9/2/3/923bd229b8d17f80eb0f929cb135141f92388534.png)
![$ AP_a,BP_b,CP_c $](http://latex.artofproblemsolving.com/0/0/4/004a51c1884d252e97029c542886cfe5f99a2d70.png)
![$ AP^*_a,BP^*_b,CP^*_c $](http://latex.artofproblemsolving.com/4/d/4/4d4478fd245dfb1a0130a1253f368237de42c9b9.png)
![$ O $](http://latex.artofproblemsolving.com/e/c/8/ec8d3f47ddcf7704bb35aa15f8eaade01a897c5e.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
![$ \Delta OP_bP_c,\Delta OP_cP_a,\Delta OP_aP_b $](http://latex.artofproblemsolving.com/b/c/4/bc44642b092e90c7c3e1efb15abb7d608fa3ceb6.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
![$ \Delta OP_bP_c,\Delta OP_cP_a,\Delta OP_aP_b $](http://latex.artofproblemsolving.com/b/c/4/bc44642b092e90c7c3e1efb15abb7d608fa3ceb6.png)
![$ A,B,C $](http://latex.artofproblemsolving.com/e/a/0/ea0e0461ab301582db5a4dab58872f4c538e3750.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ P^* $](http://latex.artofproblemsolving.com/8/5/8/8580aeee03c933fbc207886c59ce2bfa48aa71cc.png)
![$ P_bP_c $](http://latex.artofproblemsolving.com/7/6/e/76e0a8499da63d2af7ee7be3ddab62eb5a259102.png)
![$ P^*_b,P^*_c $](http://latex.artofproblemsolving.com/4/0/e/40e8a0052c1ba27bd58b566ae7797b63ddab567a.png)
![$ \angle P_bOP_c $](http://latex.artofproblemsolving.com/f/3/4/f34e8e2db10d86fecc98a63a06a42fc52c71f189.png)
![$ \Delta OP_bP_c $](http://latex.artofproblemsolving.com/0/8/c/08c0d25da093000403d630050b26781761620d28.png)
![$ \Delta OP^*_bP^*_c $](http://latex.artofproblemsolving.com/8/1/2/812f0a55672d2324effa557cd4020c025fdeb93d.png)
![$ \angle P_bOP_c $](http://latex.artofproblemsolving.com/f/3/4/f34e8e2db10d86fecc98a63a06a42fc52c71f189.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ \Delta OP_bP_c,\Delta OP_cP_a,\Delta OP_aP_b $](http://latex.artofproblemsolving.com/b/c/4/bc44642b092e90c7c3e1efb15abb7d608fa3ceb6.png)
![$ \Delta P_aP_bP_c $](http://latex.artofproblemsolving.com/9/2/3/923bd229b8d17f80eb0f929cb135141f92388534.png)
![$ AQ,BQ,CQ $](http://latex.artofproblemsolving.com/9/5/1/9512ec764dd7ca12037b6e9a51a5e068adea0409.png)
![$ P_a,P_b,P_c $](http://latex.artofproblemsolving.com/e/b/8/eb80a492ee9839d23c2593280a2b361444858372.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ \Gamma(ABC) $](http://latex.artofproblemsolving.com/b/a/f/baf37727ab229844671be67a271be09fa1c76301.png)
![$ \Delta P_aP_bP_c $](http://latex.artofproblemsolving.com/9/2/3/923bd229b8d17f80eb0f929cb135141f92388534.png)
![$ A,B,C,H,P $](http://latex.artofproblemsolving.com/1/c/7/1c77de2664d25c614132b1bc3ab51008a939dd2b.png)
![$ H $](http://latex.artofproblemsolving.com/b/2/1/b21b6cf85d4bdfe12f052f6d1451af9150b5ed8d.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ OP^* $](http://latex.artofproblemsolving.com/c/6/c/c6c53ee012bb891d66edd208215ec9f9c15a4db5.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ OP $](http://latex.artofproblemsolving.com/7/b/d/7bda632508deee45dc4c44e740a68eca66e5179a.png)
![$ Q=OP\cap HP^* $](http://latex.artofproblemsolving.com/2/2/4/2240533cb0aff8a40814e8ea67cdf7e61057c1c0.png)
![$ Q^*=OP^*\cap HP $](http://latex.artofproblemsolving.com/d/1/2/d12c0102069648235429c0338241d2f6ec299fde.png)
![$ R $](http://latex.artofproblemsolving.com/5/7/a/57aa5a5f3fbf545d11381027ea98df5fe14d4fd0.png)
![$ AP_a,BP_b,CP_c $](http://latex.artofproblemsolving.com/0/0/4/004a51c1884d252e97029c542886cfe5f99a2d70.png)
![$ R^* $](http://latex.artofproblemsolving.com/c/e/6/ce631ca5598204ae8c54526a987336d800f08785.png)
![$ AP^*_a,BP^*_b,CP^*_c $](http://latex.artofproblemsolving.com/4/d/4/4d4478fd245dfb1a0130a1253f368237de42c9b9.png)
![$ R=OP\cap HP^* $](http://latex.artofproblemsolving.com/e/9/e/e9ea3f6c3528e69d0f2f8413ede2416b8b91333f.png)
![$ R^*=OP^*\cap HP $](http://latex.artofproblemsolving.com/d/d/b/ddb20a379537089808e3b81d3cabecf2f278b3b1.png)
![$ R\equiv Q $](http://latex.artofproblemsolving.com/7/4/3/74399b473856bc64335690d2917bddde5aa3d567.png)
![$ R^*\equiv Q^* $](http://latex.artofproblemsolving.com/a/5/4/a5400eac924e1df531c436f28b520a8495ef9364.png)
![$ PQ $](http://latex.artofproblemsolving.com/5/9/6/59667f482dbe426f5c691c03076590d756f53cf9.png)
![$ \Delta P_aP_bP_c $](http://latex.artofproblemsolving.com/9/2/3/923bd229b8d17f80eb0f929cb135141f92388534.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ OP $](http://latex.artofproblemsolving.com/7/b/d/7bda632508deee45dc4c44e740a68eca66e5179a.png)
![$ O $](http://latex.artofproblemsolving.com/e/c/8/ec8d3f47ddcf7704bb35aa15f8eaade01a897c5e.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P_aP_bP_c $](http://latex.artofproblemsolving.com/8/7/0/870ab4e8251fa2fb70e714927e7101d422095297.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ P_aP_bP_c $](http://latex.artofproblemsolving.com/8/7/0/870ab4e8251fa2fb70e714927e7101d422095297.png)
![$ P_aP_bP_c $](http://latex.artofproblemsolving.com/8/7/0/870ab4e8251fa2fb70e714927e7101d422095297.png)
The proof may look long, but the idea is very small. You can easily realize that after reading the proof.
Property 4:-
If
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A $](http://latex.artofproblemsolving.com/a/3/8/a3838cb5b094421f0ff8b8f25affde76a00ddde6.png)
![$ BPC $](http://latex.artofproblemsolving.com/e/f/4/ef438bddf61934d8314278b0cb669e3d178750d5.png)
Proof:-
Note that, we have proved if the Euler lines of
![$ PBC,PCA,PAB $](http://latex.artofproblemsolving.com/5/2/1/521c253246363bdffa53bf31242a5f17addf6aab.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A $](http://latex.artofproblemsolving.com/a/3/8/a3838cb5b094421f0ff8b8f25affde76a00ddde6.png)
![$ PBC $](http://latex.artofproblemsolving.com/8/e/9/8e96ea658fe7624ee0bf8e87adc48932b73f96d4.png)
Property 5:-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ A_1,B_1,C_1 $](http://latex.artofproblemsolving.com/b/e/7/be7834f6f1cec71c27289daae4e4804e84c21551.png)
![$ AP,BP,CP $](http://latex.artofproblemsolving.com/c/3/5/c35ccd0b0e823a60baeae5ca4f323681941102cd.png)
![$ \odot PBC, \odot PCA, \odot PAB $](http://latex.artofproblemsolving.com/6/e/5/6e5405104fd2d736c68a03e5223b817e9a258162.png)
![$ H_a,H_b,H_c $](http://latex.artofproblemsolving.com/f/7/d/f7de8be5da4d509bf6ce0371973f2e9a396a0b05.png)
![$ \Delta A_1BC, \Delta B_1CA, \Delta C_1AB $](http://latex.artofproblemsolving.com/5/3/9/53993482629c26f3cc8f81567c2e3798dc31ea32.png)
![$ AH_a,BH_b,CH_c $](http://latex.artofproblemsolving.com/f/7/7/f77a24ad41a1dc6cea2feba07616a16eba3282fc.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ \Delta ABC $](http://latex.artofproblemsolving.com/e/b/a/ebac0503fd945154b26b17b5bbd9bbf7fda41540.png)
![$ BC,CA,AB $](http://latex.artofproblemsolving.com/c/0/9/c09f0577d8be2f0c4f47a4ff32b47895bcf36b80.png)
Proof:-
Suppose,
![$ O_a,O_b,O_c $](http://latex.artofproblemsolving.com/d/7/d/d7dc14fdc01649e2946902cc218c3c49265387a3.png)
![$ PBC,PCA,PAB $](http://latex.artofproblemsolving.com/5/2/1/521c253246363bdffa53bf31242a5f17addf6aab.png)
![$ CO_a,CH_b $](http://latex.artofproblemsolving.com/a/9/2/a92d0e9ae6ac34bbd56c256bf12ed3076a5bdbe7.png)
![$ \angle ACB $](http://latex.artofproblemsolving.com/7/d/2/7d271964a51b3bc5fca3962d9b38b4b8121a0694.png)
![$ A'=BH_c\cap CH_b $](http://latex.artofproblemsolving.com/1/2/5/125a72cb1f8a640095510bf3282e8b6a428c9fcd.png)
![$ B',C' $](http://latex.artofproblemsolving.com/7/b/c/7bccc52a1b418147d6b6b41f4378b16d69e33ef0.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ AO_a,BO_b,CO_c $](http://latex.artofproblemsolving.com/b/a/9/ba9e380be60ddc277cbebd554a595f877a7b0917.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ Q $](http://latex.artofproblemsolving.com/a/2/4/a24472a7c260734804c19054e87966a1645b986b.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ AA',BB',CC' $](http://latex.artofproblemsolving.com/a/7/4/a7450e1fe9fa12fcbb6bf94b246016faf2836db3.png)
![$ Q^* $](http://latex.artofproblemsolving.com/8/f/2/8f211592942557265989415ebae4d4b3ba28834f.png)
![$ BA'CB'AC' $](http://latex.artofproblemsolving.com/b/0/4/b048bfe2b841cf48a7ac06c7fc9520a3f4b462c7.png)
![$BH_a,H_aC,CH_b, H_bA,AH_c,H_cB $](http://latex.artofproblemsolving.com/8/3/8/838229d22ab5dbbb9d4a9e78b2becf7dcaef5c15.png)
![$ AH_cBH_aCH_b $](http://latex.artofproblemsolving.com/2/a/e/2ae26df8684e031405ece15d890f69e1efb139b9.png)
![$ AH_a,BH_b,CH_c $](http://latex.artofproblemsolving.com/f/7/7/f77a24ad41a1dc6cea2feba07616a16eba3282fc.png)
The other two cases are just special cases which can be verified easily.
Property 6(Concurrency of Brocard axes):-
Given a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ PBC,PCA,PAB $](http://latex.artofproblemsolving.com/5/2/1/521c253246363bdffa53bf31242a5f17addf6aab.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Note that, the last two cases are trivial cases, so we will prove the first case only.
Proof:-
Before proving the problem we will prove two lemmas.
Lemma 1:-
Isotomic line of the Leomine axis of a triangle
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
Proof:-
Suppose,
![$ A'B'C' $](http://latex.artofproblemsolving.com/c/3/2/c32e9e3fe461f06e4c97b393863b44e056c7538f.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ AA',BB',CC' $](http://latex.artofproblemsolving.com/a/7/4/a7450e1fe9fa12fcbb6bf94b246016faf2836db3.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ H $](http://latex.artofproblemsolving.com/b/2/1/b21b6cf85d4bdfe12f052f6d1451af9150b5ed8d.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ O $](http://latex.artofproblemsolving.com/e/c/8/ec8d3f47ddcf7704bb35aa15f8eaade01a897c5e.png)
![$ OH $](http://latex.artofproblemsolving.com/4/b/b/4bbea35787a833fbe8703a16d0809f5127309bd6.png)
![$ AA',BB',CC' $](http://latex.artofproblemsolving.com/a/7/4/a7450e1fe9fa12fcbb6bf94b246016faf2836db3.png)
![$ A'B'C' $](http://latex.artofproblemsolving.com/c/3/2/c32e9e3fe461f06e4c97b393863b44e056c7538f.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ A'B'C' $](http://latex.artofproblemsolving.com/c/3/2/c32e9e3fe461f06e4c97b393863b44e056c7538f.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ OH $](http://latex.artofproblemsolving.com/4/b/b/4bbea35787a833fbe8703a16d0809f5127309bd6.png)
Lemma 2:-
Let
![$ABC$](http://latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png)
![$A_1B_1C_1$](http://latex.artofproblemsolving.com/9/b/1/9b1815ca31a0a4b18b725b690862a84de67aed84.png)
![$A_2B_2C_2$](http://latex.artofproblemsolving.com/f/e/6/fe6edddc592d080ea45def0d4a92efa0d64185c1.png)
![$A_1,A_2\in BC$](http://latex.artofproblemsolving.com/a/7/0/a70f080a5e70c3c179a3e2d9bb483bf7d96fbe86.png)
![$L$](http://latex.artofproblemsolving.com/8/5/9/859ccf4cd60c7bc6b8fa1afc9a42dc811a826d6f.png)
![$A$](http://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png)
![$BC$](http://latex.artofproblemsolving.com/6/c/5/6c52a41dcbd739f1d026c5d4f181438b75b76976.png)
![$A_1B_1C_1$](http://latex.artofproblemsolving.com/9/b/1/9b1815ca31a0a4b18b725b690862a84de67aed84.png)
![$ \frac{B_1L}{C_1L}=\frac{A_2B_2}{A_2C_2}$](http://latex.artofproblemsolving.com/c/0/5/c05be1d7d76db55dc3752ac37498e06ee303d190.png)
Proof:-
Note that,
![$ \frac {B_1L}{C_1L}=\frac {AB_1}{AC_1}.\frac {AB}{AC} $](http://latex.artofproblemsolving.com/e/c/9/ec90bc2b514f9c6fd523be7d46994212123e3a0a.png)
![$ \frac {B_2A_2}{A_2C_2}=\frac {B_2C}{C_2B}.\frac {AB}{AC} $](http://latex.artofproblemsolving.com/9/3/4/9344b5b49fe60dcbc29a892d1c8f6b758feb33e2.png)
![$ B_1,B_2 $](http://latex.artofproblemsolving.com/2/6/8/2689c847ace59a2229835690c5bf604ce080fcbd.png)
![$ A $](http://latex.artofproblemsolving.com/a/3/8/a3838cb5b094421f0ff8b8f25affde76a00ddde6.png)
![$ AB_1=CB_2 $](http://latex.artofproblemsolving.com/f/f/a/ffabaf619456ca0851e5b5e589a346f8ed670268.png)
![$ AC_1=BC_2 $](http://latex.artofproblemsolving.com/d/0/1/d011dfb5a67f017e6cf3ec4a582747c73f179b03.png)
![$ \frac {B_1L}{LC_1}=\frac {B_2A_2}{A_2C_2} $](http://latex.artofproblemsolving.com/7/9/4/794ce4fe8cca1bf2bda1fe54c5cf14973ec5e316.png)
Back to Main problem:-
Suppose,
![$ O_a,H_a,K_a $](http://latex.artofproblemsolving.com/3/b/4/3b4307d5398798f975edfaa8a330b2e9b33ab29c.png)
![$ PBC$](http://latex.artofproblemsolving.com/3/2/4/324990e19c906f677e7e948e5139a747c65796c6.png)
![$ O $](http://latex.artofproblemsolving.com/e/c/8/ec8d3f47ddcf7704bb35aa15f8eaade01a897c5e.png)
![$ ABC $](http://latex.artofproblemsolving.com/7/f/d/7fdfddafd5e9aa437c94746dc414c0d97066bdd3.png)
![$ \prod_{cyc} (O_aO_b,O_aO_c;O_aP,O_aK_a)=1 $](http://latex.artofproblemsolving.com/b/5/e/b5e230344b2df41b0e741a1b4f3870dddf6d4759.png)
![$ P'B'C' $](http://latex.artofproblemsolving.com/f/7/f/f7f45d43dd607cbf3052f8d5ca4bf4675da56d21.png)
![$ PBC $](http://latex.artofproblemsolving.com/8/e/9/8e96ea658fe7624ee0bf8e87adc48932b73f96d4.png)
![$ (O_aO_b,O_aO_c;O_aP,O_aK_a)=(PC,PB;PP',B'C')=\frac {C'P'}{B'P'} $](http://latex.artofproblemsolving.com/9/f/f/9ff31356699702495cd93a1d518a2c7fcd7c200e.png)
![$ P_1B_1C_1 $](http://latex.artofproblemsolving.com/5/0/0/5004703388570d2d7b4c2508915d7874b655d1c5.png)
![$ P'B'C' $](http://latex.artofproblemsolving.com/f/7/f/f7f45d43dd607cbf3052f8d5ca4bf4675da56d21.png)
![$ P $](http://latex.artofproblemsolving.com/8/4/a/84aee4c1a5928cbb21510c2207958cc8f5d94884.png)
![$ BC $](http://latex.artofproblemsolving.com/d/1/b/d1b5a0ce1b62bd3c620eec70e59202f9f665b7e3.png)
![$ B_1C_1 $](http://latex.artofproblemsolving.com/0/8/c/08c9bac564cebd75092ec217669410332c417ee4.png)
![$ U $](http://latex.artofproblemsolving.com/f/6/f/f6fb1fbd50b66426b80c6ffae0ba4f05410ad323.png)
![$ \frac {B_1U}{C_1U}=\frac {B'P'}{P'C'} $](http://latex.artofproblemsolving.com/d/5/1/d51baed2ba81183b184d5f00ce718aae599ca46d.png)
![$ (PB,PC;PU,B_1C_1)=(PB,PC;PP',B'C') $](http://latex.artofproblemsolving.com/8/4/4/8449a325f5141daf6bd64effc0364434b88e04e3.png)
![$ (PB,PC;PU,B_1C_1)=(O_aO_c,O_aO_b;O_aO,O_aH_a) $](http://latex.artofproblemsolving.com/9/4/8/9486cd85affc659243fce2fedd39024db689058c.png)
![$ (O_aO_c,O_aO_b;O_aO,O_aH_a)=(O_aO_c,O_aO_b;O_aP,O_aK_a) $](http://latex.artofproblemsolving.com/8/1/9/819249b61baaf1433d510f3262d637b2895f7ef6.png)
![$ O_aH_a $](http://latex.artofproblemsolving.com/d/0/9/d0943207ebbb81a6f9ede5edb22f5c8ded006793.png)
![$ \prod_{cyclic} (O_aO_c,O_aO_b;O_aO,O_aH_a)=1 $](http://latex.artofproblemsolving.com/d/9/9/d998d5a7bc2cc9e310bd4a5d1b9a7915a3d62102.png)
This post has been edited 8 times. Last edited by levans, Jul 10, 2015, 5:04 PM