Question

How would this circle proof be set up?

Answer 1

Explanation:

Given: Circle O with BT as a tangent at point B.

T is the midpoint of arc CD.

To prove: (CA)(TB) = (TA)(CT)

Since point T is the midpoint of arc CD,

Therefore, m(arc CT) = m(arc TD)

and m∠CAT ≅ m∠DAT

m(∠ACT) = 90° [Angle subtended by the diameter]

By applying tangent rule in triangles ACT and ABT,

tan(∠CAT) =
`(CT)/(CA)`

Similarly, tan(∠BAT) =
`(TB)/(TA)`

Since tan(∠CAT) = tan(∠BAT)

Therefore,
`(CT)/(CA)=(TB)/(TA)`

(CA) × (TB) = (CT) × (TA)

Hence proved.