The Ungerecht Family Tree

urp:circles

Circles

• chord has two endpoints on a circle
• secant is a line that contains a chord, but extends beyond the circle
• tangent is a line that touches a circle at one point (a secant with chord length of zero)
• Intercepted arc is the part of the circle contained within the two lines
• Central angle is angle of two lines from the center of the circle, it has the same degrees as the intercepted arc
• Inscribed/Interior angle has two points and the vertex all on the circle itself, it has 1/2 the degrees of the intercepted arc

In the image, P is center point of the circle, line C is a chord and line S is a secant.

intersecting chord theorem - If two chords AB, CD intersect at P (not necessarily the center), then AP * PB = CP * PD

All inscribed angles going to the two same points on the circle (but from different vertices) have the same angle.

The intersecting secants theorem, for secants or tangents that intersect outside of the circle says the angle formed by the secant intersection = (opposite arc - adjacent arc)/2

coterminal angle - the rest of the circle outside the angle.

```eg. angle of 30°, the coterminal is 330°
in radians, use the absolute value.
2π - |angle|, or |angle| - 2π for the negative angle```

area of an arc (from a central angle):

```a = rad*r²/2  .... comes from a = πr² for full circle, and the
proportion of a circle in the arc is rad/2π

length of arc, given length of chord and radius:

```d = 2*r*sin(a/2r)
a = length of arc, d = length of chord, r = radius
```

circumcircle - a circle that touches the vertices of an polygon

``` circumcircle of a triangle - has the center at the point where the
perpendicular bisectors of the triangle's sides meet.
(might be outside the triangle).