Triangle Post
by math_explorer, May 24, 2011, 10:13 AM
(Obsolete, now that BigSams made this wonderful thread + PDF.)
For probably all triangles
:
common centers:
: circumcenter
: orthocenter
: centroid
: incenter
: excenters
side lengths:
substitution:
, 
, 
triangle inequality turns into the fact that
are positive (zero if degenerate but w/e)
common variables:
: area (sometimes 
or 
)
: semiperimeter, or half of perimeter, 
(sometimes 
)
: inradius, radius of incircle
: circumradius, radius of circumcircle (also twice radius of 9-point circle)
: exradii, radii of excircles
Law of Sines:
Area:
Heron's:
inradius-semiperimeter (by splitting into
, 
, 
) 
therefore,
corollary:
exradius-semiperimeter: analogously
therefore
a.s.o.
Euler's formula:
, 
is distance between circumcenter and incenter
Euler's inequality (implied by above):
, equality iff is equilateral
Law of Cosines:
bashing yields
distances to orthocenter:
a.s.o.
Erdos-Mordell: twice the sum of distances from
to each side 
the sum of distances from to each vertex; equality iff is equilateral and is its center
Hadwiger-Finsler:
Weitzenbock (obv. weaker):
(equality in both cases iff is equilateral)
Half-angle trig (simply use
as hypotenuse, either side as adjacent side)
opposite:, adjacent: 
, hypotenuse: 
corollary:
For probably all triangles
:common centers:
: circumcenter
: orthocenter
: centroid
: incenter
: excentersside lengths:
substitution:
, 
, 
triangle inequality turns into the fact that
are positive (zero if degenerate but w/e)common variables:
: area (sometimes 
or 
)
: semiperimeter, or half of perimeter, 
(sometimes 
)
: inradius, radius of incircle
: circumradius, radius of circumcircle (also twice radius of 9-point circle)
: exradii, radii of excirclesLaw of Sines:
Area:
Heron's:
inradius-semiperimeter (by splitting into
, 
, 
) 
therefore,
corollary:
exradius-semiperimeter: analogously
therefore
a.s.o.Euler's formula:
, 
is distance between circumcenter and incenterEuler's inequality (implied by above):
, equality iff is equilateralLaw of Cosines:
bashing yields
distances to orthocenter:
a.s.o.Erdos-Mordell: twice the sum of distances from
to each side 
the sum of distances from to each vertex; equality iff is equilateral and is its centerHadwiger-Finsler:
Weitzenbock (obv. weaker):
(equality in both cases iff
is equilateral)Half-angle trig (simply use
as hypotenuse, either side as adjacent side)opposite:
, adjacent: 
, hypotenuse: 
corollary:
This post has been edited 9 times. Last edited by math_explorer, Nov 5, 2016, 10:56 PM
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