Circumscribed triangle and tangent

Question

In the given diagram , B C T are on circle and AT tangent to the circle at T , if AB =3 and BC=4 find $${AB +AT\over AT+AC}$$

Answer 1

If you know the power of a point, then we have (with respect to point $A$ and a given circle): $$AT^2=AB\cdot AC = 3\cdot 7 =21$$ so $${AB+AT\over AC+AT}=...$$

Answer 2

Angles $ACT$ and $BTA$ both measure half the intercepted arc $TB$. So the triangles vlcontaining these angles, both also containing angle $TCA$, are similar. From the proportionality of corresponding sides of similar triangles, conclude that $|AT|$ is the geometric mean of $|AB|$ and $|AC|$. Render $|AC|=7$ from betweenness and the rest is just substituting the numbers.

Answer 3

My glitch David K, the key thing i saw that turns out to be ' power of a point' was similar triangles ATB & ATC ( along with THC ) gives side AT/ AB=AC/AT