Another Circle Theorem?

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1 Answer
Tom M.
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

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