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Circle Tangents

There are four Circles that touch all the sides of a given Triangle. These are all touched by the Circle through the intersection of the Angle Bisectors of the Triangle, known as the Nine-Point Circle.


Given the above figure, $GE=FH$, since


Because $AB=CD$, it follows that $GE=FH$.


The line tangent to a Circle of Radius $a$ centered at $(x,y)$

x'&=&x+a\cos t\\
y'&=&y+a\sin t

through $(0,0)$ can be found by solving the equation

\left[{\matrix{x+a\cos t\cr y+a\sin t\cr}}\right]\cdot\left[{\matrix{a\cos t\cr a\sin t\cr}}\right]=0,


t=\pm\cos^{-1}\left({{-ax\pm y\sqrt{x^2+y^2-a^2}\over x^2+y^2}}\right).

Two of these four solutions give tangent lines, as illustrated above.

See also Kissing Circles Problem, Miquel Point, Monge's Problem, Pedal Circle, Tangent Line, Triangle


Dixon, R. Mathographics. New York: Dover, p. 21, 1991.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 4-5, 1991.

© 1996-9 Eric W. Weisstein