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### Triangle Construction

```
Date: 09/09/2001 at 23:24:00
From: becca
Subject: Geometry-triangles

Given a triangle ABC and point D somewhere on the triangle (not a
midpoint or vertex), I am having trouble constructing a line that
bisects the area.

I have been able to construct a line parallel to one side, but it must
be at a constant ratio (1/SQRT 2). D can be anywhere. Using Sine
ratios and the formula for the area of the triangle, I have discovered
that, if D lies on BC and F is the point at which the line intersects
AB, then BA/BF = 2BD/BC. (This assumes that D intersects AB, but a
similar ratio can be derived if it intersects AC.) But I don't see how
this can help me to construct the line.

I've been working on this for days. I hope you can help!
Many thanks.
```

```
Date: 09/10/2001 at 04:05:20
From: Doctor Floor
Subject: Re: Geometry-triangles

Hi, Becca,

Thanks for writing.

C
/ \
/   \
/     E
/       \
A--D------B
x     y

I have located D on side AB, and lengths AD = x, DB = y with x < y.
In this case E must be on side BC. If x = y, then point E coincides
with C, and when x > y, E should lie on AC.

Now the region ADEC can be divided into two triangles by segment DC.

x
--- * K
x+y

where K is the area of ABC. This can be seen by noticing that the
heights of triangles ADC and DBC are equal, and their bases are
x and y respectively.

The areas of ADEC and DBE should be

1     (x+y)/2
- K = -------
2       x+y

and thus triangle DEC should have area

(y-x)/2
------- .
x+y

So the ratio of the areas of triangles DBE and DEC should be
x + y : y - x. Then, again because these triangles have equal heights
seen from bases DE and EC, this should give us that
BE : EC = x + y : y - x.

Now point A can be constructed in the following way:

C
\
E   AE//XC
\
\
X-------A----------------B
y-x       x+y

Explanation:

It is not too difficult to find point X and (since XB  is 2y, point X
is B reflected through D). Draw XC and let E be the intersection of BC
and the line through A parallel to XC.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Constructions
High School Geometry
High School Triangles and Other Polygons

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