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### Constructing a Triangle Given the Medians

```
Date: 01/01/2001 at 14:32:24
From: Ali
Subject: Geometry

Dr. Math:

How can I construct a triangle ABC given AM, BN, and CP, the
respective medians from the vertices A, B, and C?

For construction, assume the lengths, angles or whatever is given.

Can you please tell me what the construction steps are? If there are
limitations on the givens, what are those limitations?
```

```
Date: 01/03/2001 at 12:43:42
From: Doctor Rob
Subject: Re: Geometry

Thanks for writing to Ask Dr. Math, Ali.

Re-label so that AM is the longest of the three medians.

Draw line segment AM, and trisect it with points Q and R so that
AR = RQ = QM. Draw a line segment SV with length the same as that of
BN, and trisect it with points T and U so that ST = TU = UV. Draw a
line segment WZ with the same length as that of CP, and trisect it
with points X and Y so that WX = XY = YZ.

Using point R as center, draw a circle with radius ST. Using point Q
as center, draw a circle with radius YZ. Then point P will lie at one
of the two intersections of these two circles. (Either one will do.)
Draw line PQ and locate point C beyond Q on it. Connect AC. Find the
midpoint of AC, N. Draw line NQ and locate point B beyond Q on it.
Connect AB and BC.

All that is required is that the two circles intersect in two points.
That requires that the distances between their centers be less than
the sum of their radii. In symbols, that means:

QR < ST + YZ
AM/3 < BN/3 + CP/3
AM < BN + CP

This is just the triangle inequality. That means that the only
restriction on the lengths of the medians is that they themselves can
be the lengths of the sides of a triangle.

- Doctor Rob, 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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