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December 23rd, 2017, 03:17 PM   #1
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Question How to find a relationship between the elements in a circle inscribed in a triangle?

I'm stuck with this particular problem:

It states the following:

A circle is inscribed in a triangle whose $M$, $N$, and $Q$ are tangential points. The lengths of $CB=40$ and $AB=9$.

$\textrm{Find CM-AN}$

First off, I must say the segment line notation is hard for me to understand. Therefore, I assigned single letter variables such as used in algebra to the lengths which were tangential to the circle.

There is a lemma? which states the lengths from a point $P$ which are tangential to a circle have the same length therefore I assigned $CM=a$ ,$MB=b$ and $AN=c$.

The picture resulting from this is below:



As a result I figured out these relationships held:

$a+b=40$
$b+c=9$

Since what it is asked is $a-c$ it can be obtained from the previous equations as:

$a+b=40$
$-b-c=-9$

$a-c=31$

Therefore the answer should be $31$. However, I felt that the problem did not address other unknowns, which left me pondering...

As a result, I've redrawn the problem in this diagram:



I don't know if from the data provided and I'm referring to the radius of the circle, and the angles $\omega$, $\beta$ and $\phi$, and also the lengths of the lines painted with yellow, orange and navy. Can those be obtained?

I do not have much knowledge in geometry, but I recall there were two properties called Pitot and Poncelet and I don't know whether they apply in this problem and, moreover, what they are about. If somebody can help me with a redrawn diagram or hand-drawn sketch, that would be much appreciated.

If possible, can the explanations also follow the style I used? In other words, avoiding the use of segment line notation. Thanks in advance.

Last edited by skipjack; December 23rd, 2017 at 09:29 PM.
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December 23rd, 2017, 04:03 PM   #2
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By the Pythagorean Theorem, $\displaystyle AC=41$. This means that we have

$\displaystyle a+b=40$
$\displaystyle b+c=9$ and
$\displaystyle a+c=41$.

From all this you can get that $\displaystyle a+b+c=45$.

So, $\displaystyle a=36, b=4, c=5$.
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December 23rd, 2017, 11:32 PM   #3
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Quote:
Originally Posted by Chemist116 View Post
The picture resulting from this is below:
The problem doesn't tell you that angle ABC is a right angle, so your diagrams should not show it as a right angle. Your original method of answering the question is correct. It's also correct that there is insufficient information given to determine any of the individual angles or lengths other than those you are given (the lengths of CB and AB).

Poncelet showed that the diameter of the incircle of a right-angled triangle can be found very simply from the lengths of the sides of the triangle (by subtracting the length of the hypotenuse from the sum of the lengths of the other two sides). His result can be modified to cover the case where one angle is known, but isn't necessarily a right angle. As no angle is known in your problem, Poncelet's method doesn't help in solving it.

The "lemma" you mentioned is a standard result, but doesn't seem to have acquired a name. Pitot's theorem is an easy consequence of it, but usually stated in relation to a circle inscribed in a quadrilateral rather than a triangle. If ABCD is such a "tangential quadrilateral", the Pitot theorem states that AB + CD = BC + DA.
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