m348-assignment1 - Hint Consider the three pairs of similar...

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Math 348 (2006): Assignment #1 due Monday, July 10, 2006 1. Let 4 ABC be a triangle. Suppose that B = C . Prove that AB = AC , so the triangle is isosceles. Hint: Mimic the proof of Pappus’s Theorem but use ASA instead of SAS . 2. Show that a rectangle is a square if and only if its diagonals are per- pendicular. (Don’t forget to show both directions!) 3. Suppose that 4 ABC is a right triangle, with right angle C and hypotenuse AB . Show that the circumcentre is the midpoint of the hypotenuse, and the orthocentre is the vertex C . 4. Let 4 ABC be a triangle. Let D , E , and F be the feet of the three al- titudes dropped from A , B , and C , respectively. Use Ceva’s Theorem to give an alternate proof that the three altitudes of a triangle 4 ABC meet at a single point.
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Unformatted text preview: Hint: Consider the three pairs of similar tri-angles 4 ACD ∼ 4 BCE , 4 AFC ∼ 4 AEB , and 4 BFC ∼ 4 BDA . Be sure to argue correctly why these triangles are similar. 5. Let γ be a circle with centre O . Let A and B be two points on γ such that AB does not pass through O . The segment AB is called a chord of the circle. Show that the perpendicular bisector of AB passes through O . 6. Let P T be a chord of a circle γ . Let O be a point outside γ such that the line ←→ OT is tangent to γ . If P is any point on γ on the larger arc determined by P T , show that ∠ OTP ∼ = ∠ TPP . (See the right side of Figure 1.3c in the textbook.) 1...
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