Another Circle Theorem?

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1 Answer
Jan 10, 2018

Proof detailed below

Explanation:

I've taken your diagram throughout (thank you!)

We're going to give some names to parts of your diagram:

enter image source here
Let /_STQ=alpha

We will construct two radii between OT and OS

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Since the angle between a tangent and a radius is a right angle, /_OTP=90^@

:. /_OTS=90^@-alpha

Since OT and OS are both radii:

OS=OT

:. triangleOST" is isosceles"
:./_OST=/_OTS
=90^@-alpha

Since there are 180 degrees in a triangle;
90-alpha+90-alpha+/_SOT=180
180-2alpha+/_SOT=180
-2alpha+/_SOT=0
/_SOT=2alpha

Final diagram to make the last step clear:
enter image source here
I have used purple lines to make this last bit clear. We will use another circle theorem:
The angle at the centre is twice that at the circumference

:./_SOT=2/_TRS
2alpha=2/_TRS
/_TRS=alpha=/_STQ

/_TRS=/_STQ :.
the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment