PG(10)03 - , [ A,B ] respectively. Consider L ∈ BC- { B,C...

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Problems in Geometry (3) 1. Given a triangle ABC, let A 0 be the midpoint of [ B,C ] and consider Y CA -{ C,A } , Z AB -{ A,B } such that BY and CZ meet on AA 0 . Prove that Y Z is parallel to BC. 1 2. Given a triangle ABC, consider P BC -{ B,C } , Q CA -{ C,A } , R AB -{ A,B } such that AP, BQ, CR are concurrent. Let QR, RP, PQ meet BC, CA, AB in X, Y, Z respectively. Prove that (A) X, Y, Z are collinear. (B) AP, BY, CZ are concurrent or parallel. 3. Given a quadrilateral ABCD, consider X AB - { A,B } , Y BC - { B,C } , Z CD - { C,D } , T DA - { D,A } . (A) Prove that if X, Y, Z, T are collinear then XA XB · Y B Y C · ZC ZD · TD TA = 1 . (B) Is the converse true ? 4. Given a triangle ABC, let A 0 , B 0 , C 0 be midpoints of [ B,C ] , [ C,A ]
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Unformatted text preview: , [ A,B ] respectively. Consider L ∈ BC- { B,C } , M ∈ CA- { C,A } , N ∈ AB- { A,B } such that AL, BM, CN are concurrent. If P, Q, R are midpoints of AL BM, CN, respectively prove that PA , QB , RC are concurrent. 2 5. Prove that in a triangle ABC the altitude through A, the median through B and the internal angle bisector through C are concurrent iff sin A = cos B tan C . 1 It is sufficient to show that Y C Y A = ZB ZA ! 2 Observe that P, Q, R lie on the sides of the medial triangle A B C ....
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This note was uploaded on 11/14/2011 for the course MATH 373 taught by Professor Cemtezer during the Spring '11 term at Middle East Technical University.

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