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**Question 1: **You are a city planner. The three towns of
Kaneohe, Kailua and Maunawili have pooled their funds and want to
build a recreation center. The 3 towns are shown in the map below.
Where would you put the rec center so as to be fair to all 3
towns?

**Answer: **You want the rec center to be equally distant from
each of the 3 towns, right? The set of points equidistant from 2
given points is the perpendicular bisector of the segment joining the
2 points, so the point where the perpendicular bisectors meet will be
equidistant from all 3 towns. This point is the center of a circle
circumscribed around the triangle formed by the 3 towns. This point
is called the circumcenter of the triangle.

To learn more about applications of the Circumcenter, click on the link at the bottom of this page.

**Question 2:** You are a sculptor and have just completed a
large metal mobile. You want to hang this mobile, made of flat
triangular metal plates, in the State Capitol. Each triangular piece
will hang so that it will be suspended with the triangular surface
parallel to the ground. From what point should each piece hang?

**Answer: **In order to hang horizontally, parallel to the
ground, each triangle must hang from a point called the Centroid. The
centroid is the point where the medians of the triangle meet. If we
look at one median of a triangle, segment AD below, we can see that
this segment divides the original triangle into two triangles which
have equal bases (BD and CD) and the two triangles share an altitude
(AE). (Recall that the altitude of a triangle is a segment from a
vertex perpendicular to the opposite side, and that 2 of the
altitudes of an obtuse triangle fall outside the triangle.) This
makes the 2 triangles, ABD and ACD equal in area. They are not
necessarily congruent triangles.

If we construct all 3 medians of a triangle, we will find 6 triangles which are all equal in area, for reasons related to the discussion above. Therefore, the point where the 6 triangles meet, point G, is the point which is the center of balance or center of Gravity for the original triangle: the point around which the area is equally distributed, if the triangle is of uniform thickness. Therefore this is the point from which to hang the triangular metal plate so that the mobile will balance.

To learn more about mobiles, click on the link at the bottom of
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**Answer: **The center of the round house needs to be
equidistant from the street and the two properties lines: if we must
therefore be equidistant from the 3 sides of the triangle formed by
the 3 lines. If we consider 2 lines at a time, the set of points
equidistant from two lines is the bisector of the angle between 2
lines. So the set of points equidistant from all 3 lines is the point
where all 3 angle bisectors meet. This point is called the Incenter
of the triangle. Because it is equidistant from the 3 sides of the
triangle, if we use that distance as radius, we can construct a
circle tangent to all 3 sides of the triangle. This circle is the
outline of our lovely new round house!

(This architectural model was designed and built by Joseph, in my mechanical drawing class)

If you are interested in finding out more about other applications of the Incenter, click on the link at the bottom of this page.

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**Back to the Introduction to Journey to the
Center of a Triangle**