## Construction problem

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• Let ABC be a triangle. Construct a line intersecting CA, CB at Ca, Cb, resp, and another line intersecting BA, BC at Ba, Bc, resp., such that: (ACa = BCb = ABa
Message 1 of 4 , Sep 4 1:13 PM
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Let ABC be a triangle. Construct a line intersecting CA, CB at Ca, Cb, resp,
and another line intersecting BA, BC at Ba, Bc, resp., such that:

(ACa = BCb = ABa = CBc) AND (CaCb = BaBc)

A
/\
/ \
/ \
/ \
Ba Ca
/ \
/ \
/ \
/ \
/ \
/ \
B-------Cb----Bc-------C

Let ACa = BCb = ABa = CBc = x

Triangle (CaCbC): (CaCb)^2 = (b-x)^2 + (a-x)^2 - 2(b-x)(a-x)cosC (1)

Triangle (BaBcB): (BaBc)^2 = (c-x)^2 + (a-x)^2 - 2(c-x)(a-x)cosB (2)

CaCb = BaBc, and (1) and (2) ==>

(b-x)^2 - (c-x)^2 = 2(b-x)(a-x)cosC - 2(c-x)(a-x)cosB

and from this we get the x.

Synthetic solution??

Antreas

PS: Probably the point of intersection A' of CaCb, BaBc,
(if there is only one such real point !), and the similarly defined
ones B', C', are of some interest.
(for example: Are the lines AA', BB', CC' concurrent?)
• Dear Hyacinthians, let me give you the following construction problem. Construct the triangle ABC if given in position, the line on which the base BC is
Message 2 of 4 , Oct 5, 2001
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Dear Hyacinthians,

let me give you the following construction problem.

Construct the triangle ABC if given in position, the
line on which the base BC is located,and the feet of
the Cevians of O to the two lateral sides.

Best regards,
Martin

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• Dear all Given the first Brocard triangle of triangle ABC,how to construct the triangle ABC ? My solution has become unnecessarily long.Will anybody tell me a
Message 3 of 4 , Aug 31, 2002
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Dear all
Given the first Brocard triangle of triangle ABC,how to construct
the triangle ABC ?

My solution has become unnecessarily long.Will anybody tell me a precise
solution of it?

Yours faithfully
Atul.A.Dixit

_________________________________________________________________
• Johnson, page 280: === The solution depends onthe fact that any triangle is inversively similar to its first Brocard triangle. We locate the Brocard points of
Message 4 of 4 , Aug 31, 2002
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Johnson, page 280:

===
The solution depends onthe fact that any triangle is inversively similar to
its first Brocard triangle.
We locate the Brocard points of the given triangle, also its first Brocard
triangle; then by similarity we locate on the circumcircle of the given
triangle the Brocard points of the required triangle.
The vertices of the latter can then be found at once.
===

Is this what you did?

Regards,
Dick Klingens

::-----Original Message-----
::From: Atul Dixit [mailto:atul_dixie@...]
::Sent: Saturday, August 31, 2002 6:42 PM
::To: Hyacinthos@yahoogroups.com
::Subject: [EMHL] Construction problem
::
::
::Dear all
:: Given the first Brocard triangle of triangle ABC,how to construct
::the triangle ABC ?
::
::My solution has become unnecessarily long.Will anybody tell me a precise
::solution of it?
::
::Yours faithfully
::Atul.A.Dixit
::
::
::
::_________________________________________________________________