A triangle with this property is called self-conjugate with respect to the circle. The same fact affords a reformulation: the sides of the triangle are the polars of the opposite vertices with respect to the polar circle.
The law of sines then gives you the sides.
Since inversion maps circles onto circle, the circumcircle of ABC is inverted into the circumcircle of H aH bH c. This gives you all the angles in the triangle. The proof is immediate from (2): the inversion in the polar circle exchanges vertices of a triangle with the feet of the altitudes, so that, for example, A is mapped on H a. The applet helps grasp the following fact: the circumcircle and the 9-point circle of an obtuse triangle are inverse images of each other in the polar circle of the triangle. In the second pair of circles (drawn in magenta), one is the 9-point circle of ΔABC (the circumcircle of ΔH aH bH c) the second one is its inverse image in the green circle. (Note that the two red circles cross on the green one.) When the radius of the latter is such that the two red circles coincide, you can observe that the circles are orthogonal. One of the red circles is the circumcircle of ΔABC the second one is its inverse image in the green circle. The radius of the latter is controlled by the scrollbar at the bottom of the applet. There is also a green circle centered at the orthocenter. The applet shows a triangle with the altitudes and four (two red and two magenta) circles drawn. The polar circle is only defined for obtuse triangles. Thus the polar circle of ΔABC is centered at the orthocenter and have the radius R defined by (2) This circle is known as the polar circle of ΔABC. The endpoints of those tangents lie on a circle centered at H and perpendicular to all three circles. (1) then means that the tangents from H to the three circles are equal. If the feet of the altitudes opposite the vertices A, B, C are denoted H a, H b, H c, then the powers of the point H with respect to the three circles are equal: (1)įor an obtuse triangle, the orthocenter lies outside the triangle and outside each of the three circles. The orthocenter H of ΔABC serves as the radical center of the circles constructed on the sides of the triangle as diameters. |Activities| |Contact| |Front page| |Contents| |Geometry| Copyright © 1996-2018 Alexander Bogomolny
#Triangle with circle outside install
If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet. To tease out the finer details of the infinite complexity of the world, we can only glipse, but never grasp.This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Yet sometimes we must pick apart and deconstruct our system. Giving us perfection, at the expense of being a hollow representaion of the complexity of the world in itself. Sometimes we must reduce down the complexity of phenomena, to finite representation, we can then use as tools. With the need for a sophisticated enough mental model, to be contemplate both. With each paradigm possesing certain advantages and disadvantages, in any given situation. Yet crucially, both are necessaryĬontextually we must understand the world through both prisms, dependent up the specifics of the context to hand. The need to filter out, messy, unknowable edges, in the aid of coming up with clear and certain conclusions. To distill down clutter complexity, into workable, taxonomical systems we can utilise. In that of the triangle, the necessary reductionist tendancy we must possess. It embodies a perfection that is purely theoretical and will forever be beyond our practical reach. In the case of the circle, representing the infinite.
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