b Let ABCD be a cyclic quadrilateral such that the diagonals AC and BD are

# B let abcd be a cyclic quadrilateral such that the

This preview shows page 149 - 151 out of 245 pages. 6. ASSIGNMENTS, TESTS AND EXAMINATIONS Assignment 1 Solutions 1. Let ABC be a triangle with circumcentre O and incentre I . If O and I are the same point, prove that the triangle must be equilateral. [ 1 mark ] Proof. Let the angles of triangle ABC be 2 a , 2 b , 2 c and recall that I lies on the angle bisectors. This means that BAI = CAI = a , CBI = ABI = b , and ACI = BCI = c . A B C O A B C I a a b b c c If O and I are the same point, then AI = BI = CI and this means that triangle ABI is isosceles. Hence, BAI = ABI from which it follows that a = b . We can use the same argument to prove that b = c , so all angles of triangle ABC are equal. Therefore, triangle ABC is equilateral. 2. Let ABC be a triangle with circumcentre O and orthocentre H . Prove that ABH = CBO . [ 2 marks ] Proof. This problem is easy, because we already know that it’s possible to label every angle in the diagram involving triangle ABC and the circumcentre using only CAB = a , ABC = b and BCA = c . We also know that it’s possible to label every angle in the diagram involving triangle ABC and the orthocentre using the same angles A B C O A B C H E First, note that BC  #### You've reached the end of your free preview.

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