Download Steiner Point Construction Lab

Steiner Point Construction Lab
If your shortest network for three points is a Steiner Tree, but your three vertices do not form an
equilateral triangle, then you have to be more creative about finding the Steiner Point.
In the early 1600s Italian Evangelista Torricelli came up with a remarkably simple and elegant method for
locating a Steiner junction point inside a triangle. All you need to carry out Torricelli’s construction is a
straightedge and a compass; all you need to understand why it works is a few facts from high school
geometry. Here it goes:
Evangelista Torricelli
1608 – 1647
Chief discovery: Invented the mercury barometer
Also has fantastic moustache.
Finding the Shortest Network for Three Points
You need:
1. a SHARP pencil
2. a compass
3. a metric ruler (mm.)
Torricelli’s Construction
Given a triangle with vertices A, B, C and all angles < 120°.
Before you begin: Make sure that all the
angles in your triangle are < 120°.
STEP 1:
Choose any of the three sides (say BC) and construct an equilateral triangle (ABX) so that vertex A and
X are on opposite sides.
A
To construct an equilateral triangle
1. Use a compass to measure the distance between B and C
2. Mark this distance from B
3. Mark this distance from C
4. Intersection is point X.
5. Connect B and C to X.
B
C
STEP 2:
Circumscribe a circle around the equilateral triangle BCX.
X
To circumscribe an equilateral triangle
A
First: Find the center of the circle
1. set your compass to more than half of the distance BX
(think: about 2/3)
2. Mark the curve above and below BX from point B
B
3. DO NOT ADJUST COMPASS, Mark the curve above and
below BX from point X
4. Find the intersection points of two curves above and
below BX
5. Draw the line through the two intersection points. This
is the perpendicular bisector of BX.
X
6. Repeat steps 1-5 with segment CX.
7. The perpendicular bisector of CX and the perpendicular bisector of BX intersect at the
center of the equilateral triangle (and thus the center of the circle)
Second: Set the point of your compass on the center of the circle and the marking end on B, X,
or C. Draw the circle through B, X, and C.
A
STEP 3:
Join A and X with a line segment. This line segment intersects the
circle at the STEINER POINT.
STEP 4:
Connect A, B, and C to the Steiner Point to form the Steiner Tree, the
shortest network for A,B, and C. Measure the distance to find the
weight of the network.
In Class Example
Find Length of the Shortest Network using Torricelli’s Construction
C
B
X
C
Find the Steiner Point and the length of the shortest network for the following: