## Unlikey concurrences

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• To Jean-Louis Ayme and all at Hyacinthos Please forgive the last abortive post. It is almost two years since the original post, but there is something to add
Message 1 of 2 , Sep 3, 2005
To Jean-Louis Ayme and all at Hyacinthos

Please forgive the last abortive post.

It is almost two years since the original post, but there is

Here is the original relating to Ayme's paper in Crux Math

We have the following concurrences;

1. (interior) angle-bisector of A.
2. Perpendicular to 1. through B
3. Join of mid-points of AB, BC.
4. Join of touches of incircle with BC, CA.

1. Symmedian through A.
2. Join of feet of altitudes from A, B.
3. As 3. above.
4. Parallel thru B to tan at A to circumcircle.

In fact, there is a fifth line

Let T(X) denote the tripolar of a point X

THEOREM
Suppose that
A'B'C' is the cevian triangle of P
A"B"C" is the cevian triangle of Q
A*B*C* is the cevian triangle of R
where R is the crosspoint of P nd Q.
Then the following FIVE lines are concurrent
C'A'
B"A"
AR
BX
CY,
where
X = B*C*nT(P)
Y = B*C*nT(Q)

Ayme's examples are

P = G, X = X(7), so R = X(1),
P = G, X = X(4), so R = X(6).

P = G, X = X(8), so R = X(9),
P = G, X = X(69), so R = X(3).

So far, B' and C" are left out - has anyone an idea how to include
them?

Best wishes

Wilson
• Dear All Sorry - there is a slip in the previous message A*B*C* is the ANTIcevian triangle of R Wilson
Message 2 of 2 , Sep 3, 2005
Dear All

Sorry - there is a slip in the previous message

A*B*C* is the ANTIcevian triangle of R

Wilson

--- In Hyacinthos@yahoogroups.com, "Wilson Stothers" <wws@m...> wrote:
> To Jean-Louis Ayme and all at Hyacinthos
>
> Please forgive the last abortive post.
>
> It is almost two years since the original post, but there is
>
> Here is the original relating to Ayme's paper in Crux Math
>
> We have the following concurrences;
>
> 1. (interior) angle-bisector of A.
> 2. Perpendicular to 1. through B
> 3. Join of mid-points of AB, BC.
> 4. Join of touches of incircle with BC, CA.
>
> 1. Symmedian through A.
> 2. Join of feet of altitudes from A, B.
> 3. As 3. above.
> 4. Parallel thru B to tan at A to circumcircle.
>
> In fact, there is a fifth line
>
> Let T(X) denote the tripolar of a point X
>
> THEOREM
> Suppose that
> A'B'C' is the cevian triangle of P
> A"B"C" is the cevian triangle of Q

> A*B*C* is the cevian triangle of R *********************

> where R is the crosspoint of P nd Q.
> Then the following FIVE lines are concurrent
> C'A'
> B"A"
> AR
> BX
> CY,
> where
> X = B*C*nT(P)
> Y = B*C*nT(Q)
>
> Ayme's examples are
>
> P = G, X = X(7), so R = X(1),
> P = G, X = X(4), so R = X(6).
>