Prove that the bisectors of the 4 interior angles of a quadrilateral form a cyclic quadrilateral.

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You don't need a nice diagram, a lousy one will do just as well enter image description here

Let the angles of the original quadrilateral be $2\alpha,2\beta,2\gamma,2\delta$ so that $$\alpha+\beta+\gamma+\delta=180$$

So according to the diagram, the opposite angles in the small quadrilateral add to $180$

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Airdish
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Airdish

Updated on August 01, 2022

Comments

  • Airdish
    Airdish over 1 year

    I can't seem to draw a good diagram for this question. I tried to draw a quadrilateral and draw the angle bisectors, but they intersected to form a very small quadrilateral. Then I tried to draw a cyclic quadrilateral and extend the sides to form an external quadrilateral, but the diagram turned out shoddy. A diagram and perhaps a starting point hint would be greatly appreciated!

    • Wojowu
      Wojowu almost 8 years
      Hint: Letting the angles of original quadrilateral $\alpha,\beta,\gamma,\delta$, try to calculate the angles of small quadrilateral.
    • Airdish
      Airdish almost 8 years
      I know that much, but I need a nice diagram.
    • Wojowu
      Wojowu almost 8 years
      I'm sorry, but I can't help you with that. Try using some computer tool like GeoGebra if you want precise drawings.
  • Airdish
    Airdish almost 8 years
    yeah I just did it with a bad diagram, different one from yours. Thanks for the answer though... :)