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Math Help - Circle help needed

  1. #1
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    Circle help needed

    Please help me with these questions:
    1. If one side of a cyclic quadrilateral is produced, prove that the exterior angle, so formed is equal to the opposite interior angle of the quadrilateral.
    2. ABC is an isosceles triangle. D and E are any points of AB and AC respectively. If DE||BC (DE is parallel to BC), prove that DBCE is a cyclic quadrilateral.
    3. If AC and BD are two equal chords on the opposite sides of diameter AB of a circle with centre O, prove that AC||BD.
    4. Two equal circles intersect at A and B. Through A straight line MAN is drawn terminated by the circumferences show that BM=BN.

    Thanks in advance for your help!
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  2. #2
    MHF Contributor red_dog's Avatar
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    1) In a cyclic quadrilateral, the sum of opposite angles is 180^{\circ}.
    Let ABCD be a cyclic quadrilateral, E\in AD such that A\in (ED).
    We have A+C=180^{\circ} and \widehat{BAD}+\widehat{EAB}=180^{\circ}. Then \widehat{EAB}=C
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  3. #3
    MHF Contributor red_dog's Avatar
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    2) Let ABC be an isosceles triangle with AB=AC. Then \widehat{B}=\widehat{C}.
    If DE\parallel BC then \widehat{ADE}=\widehat{B}=\widehat{C}\Rightarrow \widehat{BDE}+\widehat{ECB}=180^{\circ}\Rightarrow DECB is cyclic.
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  4. #4
    MHF Contributor red_dog's Avatar
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    3) In triangles ACB and BDA we have:
    \widehat{ACB}=\widehat{BDA}=90^{\circ}
    AB=AB
    AC=BD
    Then \triangle ACB\equiv\triangle BDA\Rightarrow \widehat{CAB}=\widehat{ABD}\Rightarrow AC\parallel BD
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  5. #5
    MHF Contributor red_dog's Avatar
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    4) Let \omega_1,\omega_2 be the two circles, M\in\omega_1, \ N\in\omega_2.
    m(\widehat{AMB})=\frac{1}{2}m(arc(AB)) and m(\widehat{ANB})=\frac{1}{2}m(arc(AB)).
    Then \widehat{AMB}=\widehat{ANB}\Rightarrow \triangle MBN isosceles \Rightarrow MB=NB
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  6. #6
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    Too complex for me :(

    I've just learnt Circle (learnt 8 theorems so far).

    Please give simpler solutions. I am a Grade 10 student. Also please give reasons for what you are doing.

    (I figured out the first one, please only solve the others.)
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  7. #7
    Math Engineering Student
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    Actually, there're simple solutions, just take a look.
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