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Unformatted text preview: MATH 348: Assignment 2 (ﬁnal version)
July 15, 2008
1. Let the triangle ABC have circumradius R. Prove that its area is given
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) 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
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
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
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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