Geom6 jan 19

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## 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.

Ask a homework question - tutors are online