Concurrency of Triangle Medians
By
Kenneth E. Montgomery
Given
(Figure
1), we construct midpoints of the three segments and have
,
and
.
Figure 1:
The median of a triangle is defined
to be a segment whose endpoints are a vertex of a triangle and the midpoint of
the opposite side.
Figure 2:
,
with medians:
,
and
By constructing segments connecting
each vertex to the midpoint of the opposite side, we have medians:
,
and
(Figure
2). Open the file KM04Tri.gsp to construct the
midpoints and medians (Figure 2). It appears that the three medians share a
mutual intersection at P. Open the file KM04Anim.gsp
for an animation of
which demonstrates the properties of P, as the
size and shape of
is changed. For further exploration, open the file Assign04KM.gsp to observe P for your own
configuration of
or simply click on the intersection of the three
medians in your construction, from above and label the point P.
Concurrency
If three or more lines intersect at
a common point, the lines are said to be concurrent. The point of
intersection for such lines is said to be a point of concurrency. P is
the point of concurrency for the medians of a triangle, which is known as the centroid
Lemma: Parallel
lines divide all transversals in the in the same proportion.
Proof of Lemma: Construct
two lines m and n, such that
(Fig. 3).
Figure 3: Parallel Lines, m
and n
Let m and n be cut by
two transversals, p and q, such that p and q are
not parallel. Since p and q are not parallel, the two lines intersect at point
A, forming
.
Figure 4: Lines m and n cut by
transversals p and q, intersecting at A
The lines p and q
intersect m and n at points C, E, B and D.
and
since
and corresponding angles are congruent, and
, by the reflexive property. Therefore,
by AAA similarity. Corresponding sides of similar
triangles are proportional, so
and
, by same proportion. Thus, parallel lines divide all
transversals in the same proportion.Ò
Theorem:
The medians of a triangle intersect
at a point (the centroid, P) that is two thirds of the distance
from each vertex to the midpoint of the opposite side.
Proof:
Figure 3:
, with medians:
,
and
; and constructed segments
and
Beginning with triangle
, construct midpoints
,
and
. Construct medians,
,
and
. Construct midpoints
and
. Construct lines segments, parallel to
, through points G and H. By similar triangles, we
have FB = AF, so BG = GD and EC = AC,
so HC = DH. Therefore,
, since D = M(
). Take
to be a transversal of the three parallel lines,
,
and
. By the lemma,
parallel lines divide all transversals in the same proportion, so we see that
. Thus
is divided into 3 equal points and P lies
two-thirds of the distance from B to E. Likewise,
and P is two-thirds the distance from C
to F, so
is concurrent with
at point P. Without loss of generality, P
is two-thirds the distance from A to D. and the three medians are
concurrent at P.
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