Explore the Common Internal and Common External Tangents via this applet.
The angle where the radius intersects a tangent line is ALWAYS a right angle (90-degrees!). Try moving the center around and see what happens. You can also move the other points, as well!
Tangents Drawn to a Circle - when a tangent line is drawn to a circle, it will ALWAYS be perpendicular to the radius at the point of tangency.
"Party Hat" Theorem: If TWO tangent lines are drawn from the SAME external point (point C, here), then their tangent SEGMENTS will ALWAYS be congruent!
If two chords, AB & CD, are equidistant to the center of a circle (i.e. their 'apothems' are the same length), then the chords, themselves, will ALWAYS be congruent! Don't forget to click on "show a proof."
The inscribed angles (<SPQ = green & <RPQ = pink) that are formed when a chord intersects with a tangent at the point of tangency are ALWAYS SUPPLEMENTARY!
The inscribed angles (<JCB = red & <LCB = blue) that are formed when a chord intersects with a tangent at the point of tangency are ALWAYS HALF the measure of their INTERCEPTED ARCS!
If two chords are congruent, then their intercepted arcs are congruent!
A Deeper Look: If two chords are congruent, then their intercepted arcs are congruent.
If a diameter (EF, here) is perpendicular to a chord (CD, here), then it BISECTS the chord AND it's INTERCEPTED ARC (arc CD will also be bisected).!