Page 1 of 2 12 LastLast
Results 1 to 15 of 17

Math Help - Proofs of the Theorems on Circumscribable Quadrilaterals

  1. #1
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10

    Unhappy Proofs of the Theorems on Circumscribable Quadrilaterals

    Please help me to have some proofs of the theorems related to CIRCUMSCRIBABLE QUADRILATERALS.

    Here are some of the THEOREMS that needs to be proven;

    THEOREM (5): A point is on the angle bisector of the angle if and only if the points is equidistant from the sides of the angle, that is, if and only if the lengths of the perpendicular segments from the point to the sides of the angle are equal.

    THEOREM (6): The four angle bisectors of a quadrilateral are concurrent if and only if the quadrilateral is circumscribable.


    THEOREM (7): A quadrilateral is circumscribable if and only if the incircles of the two triangles formed by a diagonal are tangent to each other.

    THEOREM (8): The four sides of a circumscribable quadrilateral intersect the incircles of the two triangles formed by a diagonal of the quadrilateral to form four points, all of which are on a circle taht is concentric with the quadrilateral's inscribed circle.

    THANK YOU VERY MUCH AND I HOPE TO SEE THE PROOFS AS SOON AS POSSIBLE. THANK YOU!
    Last edited by mr fantastic; August 16th 2010 at 02:18 AM. Reason: Removed caps from title.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    What have you tried so far?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10
    I have tried THEOREM (5) but I have doubts on it.

    "if and only if" confused me with my proofs. I really don't know which comes first.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    If and only if proofs are bidirectional, meaning - you must prove both directions of the implication. Lets say a theorem is like this: Statement A if and only if Statement B. To prove it you need to prove the following two implications

    Statement A implies Statement B. (If A, then B)
    Statement B implies Statement A. (If B, then A)

    For theorem 5 you need to show that if a point is on the angle bisector, then it must be equidistant from the sides. The other thing you need to prove is that if it is equidistant from the sides, it must lie on the bisector. I will help you with the first part.

    Assume that a point P is on the bisector of some angle. You have two cases. If the angle is 180 degrees, it is a straight line. The bisector splits this angle into two, so we get 90 degrees and 90 degrees. You can see that any point on the bisector is trivially equidistant from the sides of the angle. Now, assume the angle is smaller than 180 degrees. Now follow the following drawing:



    So point P is on the bisector. The angle is chopped into two equal smaller angles (a). Then, we have put down the lines from the bisector to the sides of the angle. They hit the sides at 90 degrees. Hence the third angles in these two triangles are the same (b). There was a theorem that if you have 1 common side (AP) and both the angles (a and b) adjacent to that side correspond in the two triangles, then the triangles are the same. Since the triangles are the same, the sides BP and CP must be the same. Thus we have proved one side of the implication. Can you prove the other?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10
    I did that also. Thank you very much. I am correct if I say that I have to prove also that the point P (based on your figure) is on the angle bisector? because it's vice-versa, right?

    In proving the to perpendicular segments are equidistant to each other, I use the idea of Triangle Congruence Postulates and Pythagorean Theorem.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    Yes, you assume it is equidistant and then you need to prove it is on the bisector.

    For your second statement, yes it's ASA (angle-side-angle) from this page Congruence (geometry) - Wikipedia, the free encyclopedia.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    In theorem 6 does that mean that the circle is inscribed in the quadrilateral. If so, the proof should be straight-forward using theorem 5. However, I think it is false if the circle is on the outside of the quadrilateral!
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10
    yes it is inscribed in the quadrilateral. And it is actually in connection with the theorem (5). But I need to show the details. Can you please help me??
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    Here is a hint. You start by having a circle inside the quadrilateral. Now this tells you something about the center of the circle and some distances involved. Keep in mind thm 5. It should be straightforward once you draw it.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10
    I get it now. Thanks. Theorem (7) is quite simple but I am confused with theorem (8). can you please help me understand theorem (8)?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    I'm afraid I don't get that part. Could you draw a picture?
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10
    concentric circle maybe on the point of tangency in each side of the quadrilateral

    Last edited by jherv05; August 16th 2010 at 11:17 PM. Reason: no illustration
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10
    what would be the format to be used for me to post illustrations???
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    You can just upload a jpg and then post the link here.
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Newbie
    Joined
    Aug 2010
    From
    San Antonio, Sibulan, Negros Oriental, Philippines
    Posts
    10
    ah ok ill try again...

    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Diagonals of a Circumscribable Quadrilateral
    Posted in the Geometry Forum
    Replies: 3
    Last Post: July 27th 2011, 06:38 PM
  2. Replies: 6
    Last Post: June 20th 2011, 04:48 AM
  3. Replies: 4
    Last Post: June 16th 2011, 04:32 PM
  4. More proofs of more theorems
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: January 29th 2009, 12:01 PM
  5. Proofs of parallel planes theorems
    Posted in the Geometry Forum
    Replies: 2
    Last Post: August 7th 2006, 09:21 AM

Search Tags


/mathhelpforum @mathhelpforum