# Finding the length of a line in a circle when given the lengths of the sides of a triangle inside the circle

First and foremost, sorry if this was a terribly worded question. I'm pretty new to this and haven't quite got the hang of asking specific questions.

In the diagram below, I'd like to find out the length of $AX$ when $AB=6, AC=5,$ and $BC=9$.

Diagram

At first, I tried to draw a new triangle $AYX$ to solve the problem, but it didn't really help at all.I can't think of any other way of solving the problem, so I'd really appreciate it if someone could help me out.

PS: Sorry for the super-low quality image. I drew it in MS Paint. If anyone here knows a better way to draw math diagrams, I'd love to know.

• Are $A$ ,$B$ and $C$ centers here? – haqnatural Dec 24 '17 at 19:59
• Assuming the answer to haqnatural is "yes," from the diagram, it looks like (1) the large circle is tangent to the two small circles, and (2) the large circle and small circle intersect at points $X$ and $Y$. In general, it is not possible to satisfy both conditions simultaneously. Could you please clarify which is the case for your problem? – user7530 Dec 24 '17 at 20:02
• If $r_C,r_B$ are the radii of the circles with centers $C,B$ respectively , and $r=\overline {AX}$ is the radius of the large circle then $5=r-r_C,6=r-r_B,9=r_C+r_B$. – lulu Dec 24 '17 at 20:06

Trusting that $A,B,C$ are the centers of the apparent circles and that all apparent points of tangency are in fact points of tangency we define $r_C,r_B$ to be the radii of the circles with centers $C,B$ respectively , and $r=\overline {AX}$ to be the radius of the large circle. We see at once that $$5=r-r_C\quad \quad 6=r-r_B\quad \quad 9=r_C+r_B$$

Adding the first two equations yields $$11=2r-(r_C+r_B)=2r-9\implies \boxed {r=10}$$

• With r=10 we have r C = 5 which implies the point A is on the circle C. The diagram shows differently. – Mohammad Riazi-Kermani Dec 24 '17 at 20:35
• @MohammadRiazi-Kermani Good point. But the algebra shows that either $A$ is on the $C-$circle or the given data in inconsistent. Up to the OP. – lulu Dec 24 '17 at 20:43

We have a system of the equations:

\begin{eqnarray} r_A-r_B &=& 6\\ r_A-r_C &=& 5\\ r_B+r_C &=& 9 \end{eqnarray}

We are interested in $r_A$. If we sum all equations we get $2r_A = 20$, so $r_A =10$.

• Your solution indicates that point A is on circle C which is not the case in the diagram. – Mohammad Riazi-Kermani Dec 24 '17 at 20:37

The diagram is not consistent with the data provided. With the given data we get the radius of circle C to be 5 which locates the point A on the circle C.

• I'm very sorry if the diagram was drawn poorly. I did it in MS Paint, so as you might expect, it wasn't a very accurate depiction. Do you know of any better way to draw diagrams like these so that I can make them more accurate? – K. King Dec 24 '17 at 20:50