Math 427, Quiz 5.
Your name: 1. Let C be a point on side AB of ABC. Let Ra and Rb be circumradii of BCC and ACC correspondingly. As usual, a = |BC| and b = |AC|. Show that Ra b = Rb a.
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Math 427, Midterm 1.
Your name: 1. Prove that three angles in a triangle sum up to 180 .
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2. Give a proof of Pythagorean theorem using similar triangles.
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3. Let two circles and intersect at A and B. Let CD be a chord on . Let AC and BD intersect again
Math 427, Midterm 3. Your name: Each problem costs 20 points. 1. Suppose A, B, C and D are four points on projective plane, no three of which are colinear; and suppose A , B , C and D are another set of four points on projective plane, no three of which a
Math 427, Midterm 2.
Your name: Each problem costs 20 points. 1. Let ABC be a triangle on the plane with cirumradius R. Show that b c a = = = 2R. sin sin sin
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2. Suppose ABC is a triangle on the unit sphere. Show that the area of triangle is given by |
Math 427, Quiz 2.
Your name: 1. Let ABCD be a paralelogram (i.e. its opposite sides are parallel). Show that |AB| = |CD| and |AD| = |BC|. (Hint: The additional construction: diagonal BD)
B
C
A
D
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Math 427, Quiz 3.
Your name: 1. Two parallel lines l and m intersect a circle at points A, B and C, D corespodingly. Show that |AC| = |BD|.
m l A C D
B
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Math 427, Quiz 7.
Your name: 1. Give a triple pf real numbers (a, b, c) which are not projective coordinates of any point in P2 .
2. Which of the following pairs of points (in projective coordinates) are equal? (i) (1, -1, 1) and (-1, 1, -1) (ii) (-1, 1,
Math 427, Quiz 6.
Your name: 1. In a ABC with right angle at C and legs a = b = 1 one choose points X and Y correspondingly on legs BC and AC such that |CX| = 2/3 and CY = 1/3. Let P be a point of intersection of lines AX and BY and Z be the point of inte
Math 427, Quiz 4.
Your name: 1. Let I be incenter of ABC and be a circle which goes trough A, B, I. Let A = A be another point of intersection of and line AC. Show that |IA | = |IB|.
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Math 427, HWA 11, Solutions.
1. Let A, B, C, D be points on one line. Show that (A, B; C, D) + (A, C; B, D) = 1. Solution: Let a, b, c, d be coordinates of A, B, C, D correspondingly then (A, B; C, D) = and (A, C; B, D) = Therefore (A, B; C, D) + (A, C; B