Tangent Coins

Here are two ordinary coins, a quarter and a nickel. I am going to draw an external tangent from the quarter to the dime. I will be showing how drawing a line from the center of one of the coins (or circles in this case) to the point of tangency will form a right angle.

It is clear that the angle drawn from the center of the dime to its point of tangency (on the tangent shared with the quarter) is roughly 90 degrees, thus proving that any line drawn from the center to a point of tangency creates a 90 degree angle.

It is also seen here that the line from the point of tangency to the center creates a 90 degree angle.

(~DM5)

Common Tangents

After finding these materials in my room, I noticed that I was staring at a circles that had both internal and external common tangents. You could see both of the ways that made this true. The top picture shows that the two circles could share an internal common tangent, and the second picture below shows the circles with common external tangents. (~MC2)

Soda Pong with Tangents

While I was setting up a clean game of Soda Pong, I took a close look at the formation of the cups.  I then saw that the brims of the cups were externally tangent circles.  The cups touch at a point of tangency.  The radii of the cups are the same and they fit.  This makes it able for exactly six cups to fit around one center cup.  It was a great game that also allowed me to find my next Geometry Hunters post.(~KM5)

Cymbals with Tangents, Chords, & Secants

Over the weekend, I acquired two new cymbals and I thought I'd make use of them, other than for music.  In this first illustration, I proved that the result of multiplying two pieces of individual chords should be equal.  I simply multiplied AE x EC (the labels you can't see very well, sorry,) the first chord, and then BE x ED, the second chord, and they both came out to be 34.36 cm sq.  

In this second picture, I demonstrated that the tangent of a circle multiplied by itself, is equal to the outside piece of a secant multiplied by the entire length of the secant. I squared AB which resulted in 200.77 cm sq, and multiplied AC x AD, which resulted in 200.77 cm sq as well. This proves that the tangent squared does equal the secant length times the outer piece. (~JG2)

GIRL

At my local swimming pool, I found a women's bathroom sign with the shape of a perfect circle and was wondering if I could prove any geometric theorems with it.

On Geometer's Sketchpad, I drew a tangent of the circle and a secant to the circle. I planned to prove theorem 9-10 which states that two secants, two tangents, or a secant and a tangent follow the same equation. The angle that is formed by the intersecting lines would be equivalent to 1/2 the quantity of the major arch created minus the minor arch created.

Through GSP, I have proven that no matter what size or angle, the equation works every time.

(~WM5)

Books

This is a picture of a bunch of different symbols from books put together (Percy Jackson, Harry Potter, The Mortal Instruments, The Hunger Games, Divergent).

In this picture, there are three circles. The center one is externally tangent to both of the circles on either side of it. The vertical line in the middle of the picture is also the diameter of the center circle.

This image also has two lines that are externally tangent to the center circle (lines j and k). The right angles formed by the radii to the tangency point prove that the lines are tangent.

~CS1

Congruent Tangents

GC2

Tangents to a circle from a point are congruent. (Corollary)

I brought my camera to an ice cream parlor last week in hopes of finding an example of the corollary stated above. According to this corollary, tangents to a circle from a point are congruent. I found my ice cream cone to be a great example of this. In the picture, segments AB and AC are both tangent to the circle (scoop) at points B and C. This should assure that tangent AB is congruent to tangent AC, which was further proven when I measured the segments using GSP. 

Externally Tangent M&M's

A common external tangent is a tangent that does not intersect the segment joining the centers.

When I saw my sister arranging her M&M's on the table, I immediately noticed a pattern of externally tangent circles, which are coplanar circles that are tangent to the same line (just imagine it) and point. The M&M pattern shows externally coplanar circles -- the yellow points are the points of tangency and the tangent line that the circles intersect on. And because the circles are the same radius, the six M&M's around the inner circle fit perfectly. (~IR1)

iPhone Camera Externally Tangent

This is a picture of the camera and flashlight on an iPhone. The two circles of the flashlight and the camera are externally tangent. Tangent circles are coplanar circles that are tangent ot the same line at the same point. (~RG1)