2
$\begingroup$

So, I am solving EGMO and had a confusion on this problem;

A circle has center on the side AB of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD + BC = AB$.

My question is ;

Is it necessary that the center of circle is the midpoint of $AB$? Does it matter?

As I was able to solve most of the other problems, and from what I have heard, this is one of the easier problems so I shouldn't be having much trouble with it but I don't know why before starting the problem (at 2 in the night), I had this confusion in my mind so now I can't move past.

I am sorry if this is a dumb question, but please please clear my confusion.

$\endgroup$

2 Answers 2

2
$\begingroup$

Answer: No, it's not necessary that the center of the circle is, at the same time, the midpoint of $AB$.

For the proof, consider a point $T$ on $AB$, such that $BT=BC$, and let us employ directed angles. We will proceed to show that $AT=AD$. In fact, let $O$ be the center of the circle, and observe that $DOTC$ is a cyclic quadrilateral too: $$\angle ODC=\frac{\angle ADC}2=\frac{180^\circ-\angle CBA}2=\angle BTC$$ But then $$\angle DTA=\angle DCO=\frac{\angle DCB}2=\frac{180^\circ -\angle BAD}2\implies AT=AD$$

Notice: We did not require $O$ to be the midpoint of $AB$. And playing around (say: with geogebra or desmos) will show that such a configuration is possible.

$\endgroup$
1
  • $\begingroup$ thanks so much! $\endgroup$ Commented Feb 9, 2021 at 14:18
1
$\begingroup$

Attempting a sketch with Geogebra, the answer seems to be no, $O$ does not have to be the centre of $AB$.

enter image description here

$\endgroup$
3
  • $\begingroup$ thank you so much, may i please know how you managed to draw this? $\endgroup$ Commented Feb 9, 2021 at 14:18
  • $\begingroup$ @Aditya_math geogebra.org/classic - draw a circle round O, choose a point A and an extended diameter through O, draw tangents from A, choose a point D on one, draw tangents from D, choose a point C on one, draw tangents from C, call where that intersects the extended diameter B, draw a circle through A, C and D, adjust the points so B falls on that second circle, clean up the picture $\endgroup$
    – Henry
    Commented Feb 9, 2021 at 14:49
  • $\begingroup$ oh wow, thanks so much! i was not able to adjust B properly before $\endgroup$ Commented Feb 9, 2021 at 16:01

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .