High school geometry proof 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 $C$.