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Unformatted text preview: MATH 348: Assignment 2 (ﬁnal version)
Leonid Chindelevitch
July 15, 2008
1. Let the triangle ABC have circumradius R. Prove that its area is given
by
abc
.
S=
4R
Comment: This is an ordinary proof, so you may use the fact sheet.
2. Prove that the following statements are equivalent for the triangle ABC :
1) γ = π (= 90◦ );
2
2) C is the orthocenter of ∆ABC ;
3) the midpoint C of c is the circumcenter of ∆ABC .
Comment: Try to solve this one Euclid style; note that to prove the
equivalence of any three statements, it is enough to prove 1 =⇒ 2 =⇒
3 =⇒ 1. However, this is not necessarily the most convenient way.
3. Prove that a parallelogram is a rhombus if and only if its diagonals are
perpendicular.
Comment: For questions 3 and 4, don’t forget to prove both directions!
Use the deﬁnitions on the fact sheet.
4. Prove that a parallelogram is a rectangle if and only if its diagonals are
equal.
5. Let IA , IB , IC be the excenters of ∆ABC . Prove that the orthic triangle
of IA IB IC is ABC .
Comment: Recall that that the orthic triangle has the feet of the altitudes of the original triangle as its vertices.
6. Use Ceva’s theorem to prove the existence of the orthocenter.
Comment: You need to prove that the altitudes are concurrent using
Ceva’s theorem.
BONUS. Let O, R and I, r be the center and radius of the circumcircle and the
incircle of ∆ABC , respectively. Show that
OI 2 = R2 − 2rR. 1 ...
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This note was uploaded on 12/10/2011 for the course MATH 348 taught by Professor Karigiannis during the Summer '06 term at McGill.
 Summer '06
 Karigiannis
 Math, Geometry

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