Geometry.

1. The determinant of a $2 \times 2$ matrix:
   
   $$\left\vert \begin{array}{*{2}{c}}a & b \\ c & d\end{array}\right\vert = ad-bc.$$
   
   

2. Cramer's Rule: To solve
   
   $$ax+by = e \mbox{ and } cx+dy = f$$
   
   
   define:
   
   $$\Delta = \left\vert \begin{array}{*{2}{c}}a & b \\ c& d\end{a... ...gin{array}{*{2}{c}}a & e \\ c& f\end{array}\right\vert = af-ec.$$
   
   

   If $\Delta \neq 0$ then the answer is
   

   $$x=\frac{\Delta_x}{\Delta}, y=\frac{\Delta_y}{\Delta}.$$
   
   

   If $\Delta=0$ and at least one of $\Delta_x,\Delta_y$ is non zero then there is no solution.

   If all $\Delta, \Delta_x, \Delta_y$, then there are infinitely many solutions.

3. The distance between two points A and B on the real line is $d(A,B)=\vert A-B\vert$.
4. For two points $P_1(a_1,b_1)$ and $P_2(a_2,b_2)$ in the xy-plane:
5. The distance between the points is
   
   $$d(P,Q) = \sqrt{(a_2-a_1)^2+(b_2-b_1)^2}.$$
   
   

6. The parametric two point form of the line containing these points is
   
   $$x = a_1 + t(a_2-a_1), y = b_1 + t(b_2-b_1).$$
   
   

7. The compact parametric two point form of the line is
   
   $$(x,y) = P_1 + t(P_2-P_1) \mbox{ or } (x,y) = (1-t)P_1 + tP_2.$$
   
   

8. The midpoint of two points $A,B$ is $\frac{A+B}{2}$.

9. The two point form of the equation through the points $(a_1,b_1), (a_2,b_2)$ is
   
   $$(a_2-a-1)y - (b_2-b_1)x = a_2b_1-a_1b_2.$$
   
   

10. The slope of the line through two points $(a_1,b_1), (a_2,b_2)$ is

    
    

    $$\frac{b_2-b_1}{a_2-a_1}$$
    
    
    where the slope is infinite and the line is vertical if $a_1=a_2$.

11. The slope intercept form of the line joining $(a_1,b_1), (a_2,b_2)$ is $y = mx +c$ where $m$ is the slope and $c = \frac{a_2b_1-a_1b_2}{a_2-a_1}$ is the y-intercept.
12. If $0\neq p$ is the $x$ intercept and $0\neq q$ is the $y$-intercept, then the line is:
    
    $$\frac{x}{p} + \frac{y}{q} =1.$$
    
    

13. A line parallel to $ax+by=c$ is $ax+by=k$ for some $k$.
14. A line perpendicular to $ax+by=c$ is $bx-ay=k$ for some $k$.

15. The quadratic function $Q(x)=ax^2+bx+c$ has extremum value at $x=-\frac{b}{2a}$. The value is $\frac{4ac-b^2}{4a}$. It is a maximum if $a<0$ and minimum if $a>0$.

16. The equation $ax^2+bx+c=0$ with $a\neq 0$ has solutions:
    
    $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\leqno{\mbox{ Quadratic Formula}}.$$
    
    

17. The equation of a circle with center $(h,k)$ and radius $r$ is
    
    $$(x-h)^2+(y-k)^2=r^2.$$
    
    

18. For the circle
    
    $$x^2+y^2+ux+vy=w$$
    
    
    the center is $(-u/2,-v/2)$ and radius is $\sqrt{w+\frac{u^2}{4}+\frac{v^2}{4}}$.

19. A parametric form of the circle $x^2+y^2=r^2$ is:

    
    

    $$(x,y) = \left( r\frac{1-m^2}{1+m^2},r\frac{2m}{1+m^2}\right)$$
    
    
    where $m$ is the parameter.

    The trigonometry parameterization for the same circle is: $z=r\exp(i\theta)$ or
    

    $$(x,y)=(r\cos(\theta),r\sin(\theta))$$
    
    
    with parameter $\theta$.

20. A circle with diameter $(a_1,b_1), (a_2,b_2)$ is:
    
    $$(x-a_1)(x-a_2)+(y-b_1)(y-b_2)=0.$$
    
    

21. The distance from a point $(p,q)$ to the line $ax+by+c=0$ is:
    
    $$\frac{\vert ap+bq+c\vert}{\sqrt{a^2+b^2}}.$$
    
    

    The expression $ap+bq+c=0$ if and only if the point lies on the line.

    Moreover, the sign of the expression $ap+bq+c$ changes if the point moves from one side of the line to the other.