High school geometry proof help
I need help with a high school geometry proof. I think I've figured out why the prompt is true, but the proof attempt I've come up with seems very inelegant. Is there an easier method I'm missing?
Consider two circles with the second internally tangent to the first at point $A$ and also passing through the center of the first. Show that every chord of the first circle which has $A$ as an endpoint is bisected by the second circle.
My attempted proof:
Let there be any chord of the first circle which has $A$ as an endpoint. Let the other endpoint of the chord be called $B$.
Then let the following line segments be drawn:
- A segment connecting $A$ and the first circle's center $C$;
- A segment connecting $B$ and the first circle's center $C$; and
- A segment connecting the first circle's center $C$ with the point $I$ where the chord intersects the second circle.
Segments $AC$ and $BC$ are the same length because they both represent the radius of the first circle.
We have two right triangles $ACI$ and $BCI$. Since the two hypotenuses $AC$ and $BC$ are the same length and the two heights $CI$ are the same length, then the two bases $AI$ and $BI$ must also be the same length. Since $AI$ is half the length of the chord, the second circle bisects the chord.
Thanks in advance for your help.
Another proof: Since both CI and BD are perpendicular to AB, CI is parallel to BD. The fact that C is the midpoint of AD implies that I is the midpoint of AB.
From the tangency point $A$, blow up the small circle such that $$\text{The small circle} \rightarrow ^{\text{gets mapped to}} \text{The circle twice as big}$$
Since the dilation is of factor two, QED.
Though what I've said is a mere restatement of what others said before, the idea of dilation, called Homothety in general is an useful tool for Eucledian Geometry (check eg. the proof of the existence of Nine Point circle)
Another proof is to take the center of the circles being $C$ (the bigger), $D$ the smaller, the intersection point $I$ for the bigger circle and $J$ for the smaller.
We have that the triangles $ADJ$ and $ACI$ are similar since they have congruent angles (which is because they are isosceles and sharing an angle (of the leg). Also we have that $AD$ is half of $AC$ which implies that $AJ$ is half of $AI$.
We know that $D$ is on $AC$ because if it weren't we would have that $ADC$ would form a proper triangle and we would have that the smaller triangle doesn't tangent the larger. We also have that $D$ is the midpoint or otherwise the smaller circle wouldn't pass through $C$.
I think next two facts give some hints.
- In a circle chords, based on the same length arcs are of the same length
- Arcs of the same angle are proportional to circle radius.
This immediately gives you $2AI=AB$, since those are chords on the arcs of the same angle in circles with radius ratio of $1:2$.
Here is an algebraic way. Let $r$ be the radius of the small circle and $R$ be the radius of the big circle. It's obvious that $R=2r$. Now, let $AB$ be the chord of the small circle and $AC$ be the chord of the big circle. Let point $A$ be the center of coordinate $(0,0)$, point $B(x_1,y_1)$, and point $C(x_2,y_2)$. The equation of small circle is \begin{align} x_1^2+(y_1-r)^2&=r^2\\ x_1^2+y_1^2&=2ry_1\tag1 \end{align} The equation of big circle is \begin{align} x_2^2+(y_2-2r)^2&=4r^2\\ x_2^2+y_2^2&=4ry_2\tag2 \end{align} From $(1)$ and $(2)$ we have $AB^2=2ry_1$ and $AC^2=4ry_2$. Then $$\frac{AB}{AC}=\sqrt{\frac{y_1}{2y_2}}$$ It's easily noticed that $\frac{y_1}{y_2}=\frac{1}{2}$, hence $$\frac{AB}{AC}=\frac{1}{2}$$