(b) Let ABCD be a cyclic quadrilateral such that the diagonals AC and BD are perpendicular and intersect at the point P . Let M be the midpoint of AB and let N be the point on CD such that PN is perpendicular to CD . Prove that the points M , P , N lie on a line. [ 4 marks ] 8. In the hyperbolic plane, there exist pentagons all of whose angles are equal to 90 ◦ . Sketch one example of such a pentagon in the hyperbolic plane using the Poincar ´ e disk model. What is the area of this pentagon? [ 2 marks ] 9. Let D , E , F be points on sides AB , BC , CA of triangle ABC , respectively. If DE = BE and FE = CE , prove that the circumcentre of triangle ADF lies on the angle bisector of ∠ DEF . [ BONUS ] Hints 1. Draw in the line segments AI , BI , CI or equivalently, AO , BO , CO . What is special about the lengths in the diagram? What is special about the angles in the diagram? 2. We already know how to work out all the angles in the diagram with triangle ABC and circumcentre O . We also already know how to work out all the angles in the diagram with triangle ABC and orthocentre H . 3. (a) Altitudes lead to right angles and right angles often lead to cyclic quadrilaterals. You should be able to use a cyclic quadrilateral or two to obtain some useful relations between angles. (b) Equal chords in a circle subtend equal angles — but remember that equal angles are also subtended by equal chords. 4. Try to mimic the case that we covered in lectures as much as possible. 5. Since the problem involves a tangent, you should try to use the alternate segment theorem. Use this to prove that ∠ LBX = ∠ KCX . 6. How would you prove that triangle XYB is isosceles? 7. (a) Suppose that C lies outside the circle with diameter AB — can you find a contradiction? (b) You should use the result from part (a), even if you are unable to prove it. 8. Do you remember how we found the sum of the angles in a polygon with n sides? 9. Do you really expect hints for a bonus question? 147
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
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