Easier center finding technique

[mklotz]

A traditional technique to find the center of a circle is to construct two chords and then use a compass to construct the perpendicular bisectors of those chords. The center is then located where those two bisectors cross.

However, there is a simpler technique that doesn't require the compass and can be done more quickly. It utilizes an interesting bit of circle geometry...

If YOU pick a random point on the circumference of a circle and from it draw straight lines to the ends of a diameter you'll find that the vertex angle these lines form at the circumference is ALWAYS a right angle.[A proof of this follows below.]

Rather than start with the diameter and construct the angle on the circumference, let's place a right angle on the circumference and mark where its legs contact the circumference. Those two points should lie on a diameter and, if you can find two (different) diameters their crossing point is the center of the circle.



I've positioned the right angle of one of my drafting triangles on the circle and marked the three points where it contacted the circumference.



After connecting the points defined by the legs of the right angle with a straight line, we note that the line passes through the middle of the circle; therefore it is a diameter.



Repeating the process with another randomly chosen point on the circumference yields another diameter that intersects the first diameter at the center of the circle.

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In mathematics you don't really understand anything until you can prove it...



APB is termed an "inscribed angle"; its vertex lies on the circumference of the circle.

AOB is termed a "central angle"; its vertex lies at the center of the circle.

Both of these angles intersect the same arc on the circumference, AB

We want to prove that, under these conditions, the central angle is always twice as big as the inscribed angle.

By drawing the radius, OP, we create two smaller triangles, POAP and POBP

Each of these triangles are isosceles so their base angles, 'a' on the left, 'b' on the right are equal.

Then the angle POA = 180 - 2a and POB = 180 - 2b

Along with the central angle, AOB, these angles must add to a complete circle, 360 degrees.

(180 - 2a) + (180 - 2b) + AOB = 360

which reduces to:

AOB = 2 (a + b)

The inscribed angle satisfies APB = a + b

So, the central angle is always twice the inscribed angle if they both subtend the same arc on the circumference. QED

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Now we'll use what we proved above to explain how our center finding process works...

If APB = 90 then AOB = 180, i. e. a straight line, and divides the circle into two equal 180 degree halves. Thus it must be a diameter and, if it's a diameter, it passes through the center of the circle.