Geom6 jan 19 - centroid divides each median in the ratio 2...

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PEM Level C 16 January 2010 Topics in Geometry: More on Triangles Theorems: 1. The perpendicular bisectors of the sides of a triangle are concurrent. 2. (Brahmagupta’s Theorem) The product of the lengths of two sides of a triangle is equal to the product of the length of the altitude to the third side and the circumdiameter. 3. (Ceva’s Theorem) Let P,Q, and R be the feet of the cevians from vertices A,B, and C of 4 ABC . If the cevians AP,BQ, and CR are concurrent, then AR RB · BP PC · CQ QA = 1 . 4. (Converse of Ceva’s Theorem) Let P , Q , and R be the feet of the cevians from vertices A , B , and C of 4 ABC . If AR RB · BP PC · CQ QA = 1 , then the cevians AP , BQ , and CR are concurrent. 5. Let D , E , and F be the midpoints of the sides BC , AC , and AB , respectively, 4 ABC . Then (a) EF k BC,DF k AC, and DE k AB, so that 4 DEF ∼ 4 ABC ; (b) EF : BC = DF : AC = DE : AB = 1 : 2, so that the perimeter of 4 DEF is one-half that of 4 ABC ; (c) ( DEF ) : ( ABC ) = 1 : 4; and (d) 4 ABC and 4 DEF have the same centroid. 6. A triangle is dissected by its medians into six smaller triangles of equal areas. Moreover, its
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Unformatted text preview: centroid divides each median in the ratio 2 : 1. Exercises: 1. Let R be the circumradius of 4 ABC . Then abc = 4( ABC ) R ). 2. Let ABCD be a cyclic quadrilateral. Prove that AC ( AB · BC + CD · AD ) = BD ( AD · AB + BC · CD ) = 4 R ( ABCD ), where R is the radius of the circumcircle of the quadrilateral ABCD . 3. In the figure, the cevians AD , BE , and CF of 4 ABC meet at P . Given that BD : DC = 5 : 4, AE : EC = 6 : 5, and ( APF ) = 48, find ( BPF ). 4. Let M be the midpoint of the side AB of an equilateral triangle ABC . Let N be a point on BC such that MN ⊥ BC . Prove that 4 BN = BC . 5. In 4 ABC , a line m is drawn through its centroid G such that the vertices A and B are on the same side of m . Let X , Y , and Z be the feet of the perpendiculars of m from A , B , and C , respectively. Prove that AX + BY = CZ ....
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This note was uploaded on 11/15/2010 for the course MATH 100 taught by Professor B during the Spring '09 term at Ateneo de Manila University.

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