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# Is there any prove that tangents from an external point of a circle are equal in length and the perpendicular bisector of any chord pass through the centre of the circle?

Question #74663. Asked by kaung30. (Jan 18 07 2:34 AM)

Baloo55th

There won't be a proof that tangents from an external point of a circle are equal in length, because there is no definition of the length of a tangent anyway. A tangent is a line touching a circle and at a right angle to a radius of the circle. It must therefore be external to the circle, and can be of whatever length you feel inclined to give it. A chord, on the other hand, is internal to the circle, and is a straight line touching the circumference at its ends. If you draw lines from the ends of the chord to where the bisector reaches the circumference, you will have two right angled triangles. As these have two sides equal in length (part of the bisector is shared, and the other if half the chord), the two triangles are equal in size. It follows from this (to me, at least) that this means that the bisector not only bisects the chord, but also the circle and therefore must pass through the centre. The diameter of a circle is a special case of a chord that actually passes through the centre, as does its bisector. Dress this up in maths talk and you might get away with it. Gmack or peasy will probably come up with a better explanation.

 Jan 18 07, 7:43 AM
zeller

If you draw 2 tangents from an external point along with a line from the centre of the circle as well as the radii to the points where the tangents meet the circle you have created 2 right-angled triangles. Moreover, these are congruent (identical) triangles because of common length, radii and rightangle. Therefore the tangents must be equal in length.

 Jan 18 07, 4:09 PM
Baloo55th

Yes, you can draw tangents that are equal in length. You can draw lines that are different lengths, or even think of lines that are infinite. Take a circle and draw a radius. From the point where the radius touches the circumference draw a line 2 inches long at right angles to the radius and label the end A. Label the point where the radius touches the circumfeence P. From P draw another line one inch long at right angles to the radius but going the opposite way and label the end B. You now have two tangents (PA and PB) to the circle that are of unequal length. Apart from the definition I gave above which referred to circles particularly, a tangent is merely any line which touches a curve. It can be of any length, and in the case of 'wiggly' curves, a line may be tangent to the curve at more than one place, and may also intersect the curve.

 Jan 18 07, 4:38 PM

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