Triangle:
1) Area of triangle with vertcies (x1,y1), (x2,y2), (x3,y3) = 1/2[x1(y2-y3) + x2(y3-y1) + x3(y1-y2)]
2) Area of circumcircle in equilateral ∆ with side a = Pi*a^2/3
3) Area of incircle in equilateral ∆ with side a = Pi*a^2/12
4) Altitude of equilateral ∆ with side a = a*sqrt3/2
5) An equilateral triangle has the maximum area

Circle:
1) Equation of circle with center at (h,k) and radius (r) = (x-h)^2 + (y-k)^2 = r^2
2) Two circles will touch or intersect each other if the distance between their centers ‘d’ is such that R-r <= d <= R+r
3) The equation of tangent to the circle x^2 + y^2 = r^2 at the point (x1,y1) is x*x1 + y*y1 + r^2=0


Line/Linearity:
1) Distance of point (x1,y1) from line ax+by+c = |ax1+by1+c|/sqrt(a^2+b^2)
2) Reflection of a point (x,y) across a line y=x is (y,x)
3) Reflection of point (x,y) across a line y=-x is (-y,-x)
4) Coordinates of point dividing a segment (x1,y1) and x2,y2 in proportion r:s is (rx2+sx1/r+s), (ry2+sy1/r+s)
5) If the slope of line is negative, line slants downward from left to right (\)
6) If the slope of line is positive, line slants upward from left to right (/)



Misc1:
1) Number of diagonals = N(N-3)/2 ; N= number of sides
2) Area of square = 1/2 * d^2 ; d = diagonal


Number Theory:
1) Any perfect square can be expressed in the form 4n or 4n+1
2) If A:B=C:D, then A+B/A-B = C+D/C-D
3) If a/b=c/d=e/f….. then a/b=c/d=e/f=a+c+e/b+d+f
4) LCM * HCF = product of two numbers
5) Sum of first n natural numbers = n(n+1)/2
6) Sum of first n even numbers= n(n+1)
7) Sum of first n odd numbers= n^2
8) Sum of cubes of 1st n natural numbers = [n(n+1)/2]^2
9) Sum of squares of 1st n natural numbers = n(n+1)(2n+1)/6

Misc2:
1) Clock Angle = mod [(60H-11M)/2]; where H = value of hour hand, M = value of minute hand. ex. 2:30, H=2, M=30

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Let me know if they are helpful. Will add more..