Circle Theorems

Circles are arguably one of the most important shapes, not just in geometry classes, but also in the world around us. Circle theorems are a very important aspect of Euclidean geometry that has everything to do with circles and angles. Like lenses, with the right ones, that mess of lines suddenly becomes something of geometrical beauty.

Kai G

Here are some of the most common and useful circle theorems:

Circle Theorem 1

A tangent line and a radius will always intersect at 90°.

A tangent is a line that only touches a single point on the circumference of a circle. There is a simple proof for this: The shortest distance between 2 points will always be perpendicular, as any non-perpendicular pathway will result in a right-angled triangle, where the new pathway becomes the hypotenuse, which, as Pythagoras proved over 2500 years ago, is the longest side of the triangle. Therefore, the line that connects the centre of the circle to the point of intersection of the tangent and circumference, the radius, must be perpendicular to the tangent.

Circle Theorem 2

Two radii form an isosceles triangle.

This is rather intuitive, but nonetheless a very important theorem. An isosceles triangle is a triangle that has at least two sides of the same length. Since two of the sides are radii of the same circle, they must also be the same length: $OA = OB = r$. Therefore, it fulfils the requirements for an isosceles triangle.

The Greek mathematician Archimedes calculated an approximation of π by inscribing and circumscribing polygons within circles — a method that laid the foundations for modern calculus over 2000 years later. Archimedes, using 2 96-sided polygons, estimated π to be between 3 $\frac{10}{71}$ and 3 $\frac{1}{7}$ (approximately 3.1408 and 3.1429). That is correct to 3 significant figures, which was an astonishing feat considering it was over 2000 years ago.

Circle Theorem 3

The perpendicular bisector of a chord intersects (passes through) the centre of a circle.

A chord is a straight line that passes through a circle. A perpendicular bisector is a line that passes through a given line segment at its midpoint at 90°. This is not so intuitive, yet its proof is fairly satisfying and easy to follow.

Consider this diagram, where $AB$ is a chord, and $OX$ is the perpendicular bisector. Observe the triangles $AXO$ and $BXO$:

$OA = OB$ (They are both radii of the circle)

$AX = BX$ (Since $X$ is the midpoint of the chord $AB$)

$OX = OX$ (Shared side)

Therefore, by SSS (side-side-side) congruency, the two triangles are congruent.

It follows that angles $\angle OXA$ and $\angle OXB$ are equal, and since angles on a straight line sum up to 180°, $\angle OXA = \angle OXB = 90°$, thus proving the theorem.

Circle Theorem 4

The angle subtended at the centre of the circle is twice the angle subtended at the circumference from the same two points.

“Subtended” just means “formed”.

This theorem sounds a lot more complicated than it really is. Here is a visual representation of the theorem, followed by its proof.

Consider the diagram. Angle $\angle OPR$ is the angle subtended at the centre, and angle $\angle PQR$ is the angle at the circumference. $QS$ is the diameter of the circle. $\angle RQO = \angle ORQ = a$ and $\angle PQO = \angle OPQ = b$.

By the Exterior Angle Theorem, $\angle ROS=\angle RQO+\angle ORQ=a+a=2a$.

A similar argument is applied to angle $\angle POS$.

Therefore, $\angle POR=2(a+b)$ and $\angle PQR=a+b$.

It follows that $POR=2PQR$, hence the Theorem.

The German child prodigy Carl Friedrich Gauss requested a 17-sided polygon to be inscribed on his tombstone. At the age of 19, he proved that one could construct a heptadecagon while only using a protractor and a ruler. He did this by proving that $cos(\frac{2\pi}{17})$ could be constructed, as it could be written as a series of nested square roots, which can be constructed by just using a ruler and other basic mathematical principles.

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