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Exercises 4.6, #8 Solution:
It is given that M, N, and P are the midpoints of sides KB, KC, and E of AABC, respectively. To Prove: The centroid of AMNP is the same point as the centroid of AABC. Proof: By the Midpoint Connection Theorem, Theorem 4.2.15,
applied to AABC, MN M AC . By the Midpoint Connection Theorem, Theorem 4.2.15,
applied to AABC, NP ll AB . Therefore, quadrilateral AMNP is a parallelogram. By Theorem 4.2.11, the diagonals of parallelogram AMNP bisect
each other. Let R be the po_ir_1t_ where these diagonals intersect. Thus, R is the midpoint of MP . Therefore, NR is a median of AMNP . Since A  R  N , the median AN extending from vertex A of AABC anL the median ﬁ extending from vertex N of AMNP lie along the same line, AN . Recall that it was shown that MN H E . B By the Midpoint Connection Theorem, Theorem 4.2.15L
applied to AABC, MP H BC . Therefore, quadrilateral MBNP is a parallelogram. By Theorem 4.2.11, the diagonals of parallelogram MBNP bisect
each other. Let S be the point where these diagonals intersect. Thus, S is the midpoint of MN . Therefore, PS is a median of AMNP . A p C Since B  P N , the median 5 extending from vertex B of AABC andH the median RS extending from vertex P of AMNP lie along the same line, BP . Let T be the centroid of AABC. H H
Then, T is the point of intersection of the two lines AN and BP which contain two of the medians of AABC .
Since the two lines also contain two of the medians of AMNP , T is also the centroid of AMNP . Therefore, the centroid of AABC and the centroid of AMNP are both the same point T . QED §_E‘xa~c:3€s Li» 7 *9 Ba, Thaw (t.7Z)1_t\l {chivald '5 I 'p— § 3 64445an wlfzt ﬂu. ()P‘l’kocenfgh M 7Q i Giraumcenjfr‘ 4/ a fwlmogﬁ . ,. i S’tth‘I‘xLEqu/gvéane. "S TZurpmﬁcaloL lino) ..
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Exercises 4.8 Ineach ofﬂtetdanglesinExcrcises l through4, Hismcorthoccntcran
is the circumcenter. Locate the center and identify the length of the radius
the ninepoint circle associated with each triangle. Use a compass 00‘ draw the
ninepoint cimle for eachniangle. '
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 Spring '10
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