SOLUTION: Two equal chords AB and XY of a circle intersect at E, a point inside the circle. Use equal arcs to prove that AX=BY and that ∆EBX is isosceles.

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Question 1195512: Two equal chords AB and XY of a circle intersect at E, a point inside the circle. Use equal arcs to prove that AX=BY and that ∆EBX is isosceles.
Answer by ikleyn(52781) About Me  (Show Source):
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Two equal chords AB and XY of a circle intersect at E, a point inside the circle.
Use equal arcs to prove that
    (a)   AX=BY
  and that
    (b)   ∆EBX is isosceles.
~~~~~~~~~~~~~~~~ 

Make a sketch on your own, as described in the problem.


So, you have a circle and two chords AB and XY in it.
The chords intersect at the inner point E.


Since the chords are equal, the corresponding minor arcs AB and XY are congruent 
(have the same arc lengths).


Arc XB is the common part (the intersection) of minor arcs AB and XY.
When you subtract (remove) this common part (the intercetion) from arcs AB and XY,
you will get equal remaining arcs AX and BY. 

So, arcs AX and BY have equal arc lengths.
Due to the basic arcs property *), it implies than tighten chords AX and BY are congruent,
i.e. have equal lengths: AX = BY.

QED.

First statement is proved.


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*)   Here this basic property is formulated explicitly: 

    +---------------------------------------------+
    |    Two chords in a circle are congruent     |
    |             if and only if                  |
    |    their corresponding arcs are congruent.  |
    +---------------------------------------------+


You may find it in any serious Geometry textbook.  It is a part of the standard  Geometry curriculum,
which is a  PRE-REQUISITE  for solving such problems - so,  I  assume that you are familiar with it.

Otherwise,  you may find the proof of this property in the lesson
    - The longer is the chord the larger its central angle is
in this site.


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Next,  the proof of the second statement is in couple of lines. 

In triangle EXB, consider angles EXB and EBX.

These angles lean on arcs BY and AX, respectively.

But these arcs are congruent, as I proved it above.

Hence, angles EXB and EBX are congruent.

Thus, triangle EXB is isosceles, since it has congruent base angles at tyhe bas XB.

The proof and the solution is complete.


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At this site,  you have free of charge online textbook on Geometry
    GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

The referred lesson is the part of this online textbook under the topic  "Properties of circles, their chords, secants and tangents".


Save the link to this online textbook together with its description

Free of charge online textbook in GEOMETRY
https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson

to your archive and use it when it is needed.


        This textbook contains  whole/(entire)  standard high-school
        Geometry curriculum in its logical development from  " a "  to  " Z ".