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Math Help - Construction - inscribe a circle in a given triangle

  1. #1
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    Construction - inscribe a circle in a given triangle

    I just started reading George Polya's book "Mathematical Discovery", and the first chapter concerns geometric constructions using a straightedge and compass. I have a weak background in basic geometry and have a question about one of the constructions he describes. It is:

    Inscribe a circle in a given triangle.

    He first reduces the problem to the construction of the center of the required circle. Then he describes the condition of the problem as "the point X [the center of the circle] should be at the same perpendicular distance from the three sides of the given triangle". Then he splits the condition into two parts: 1, that X is equidistant from sides A and B, and 2, that X is equidistant from lines A and C. Then he concludes:

    The locus of the points satisfying the first part of the condition consists of two straight lines, perpendicular to each other: the bisectors of two of the angles included by A and B. The second locus is analogous. The two loci have four points of intersection: besides the center of the inscribed circle of the triangle we obtain also the centers of the three escribed circles.

    I've attached a drawing of what I understand by the problem. But I see five points of intersection including the center. And despite some looking on the Internet I can't find out exactly what is meant by "escribed circles" in this case. Could anyone tell me what the points of intersection are supposed to be and what the three escribed circles look like?
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  2. #2
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    The attached diagram (taken from this Wikipedia page) shows what Pólya means. The red lines are the "internal bisectors" of the angles of the triangle ABC. The green lines (each green line is perpendicular to the red line through the same vertex) are the "external bisectors" of those angles. The blue circle is the inscribed circle. The three yellow circles are the escribed circles. Each escribed circle touches one side of the triangle internally, and the other two sides externally (in other words, when they are extended beyond the original triangle).

    The four points of intersection are the points where three of the red or green lines intersect. These points are I, the incentre, and J_A,\: J_B,\: J_C, the three excentres.
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  3. #3
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    Thank you so much! I guess I didn't realize they were called excircles. Out of curiosity though: Polya doesn't mention what the radius of the incircle is. There is a formula for it in the Wikipedia article, but it seems a little complicated and I was wondering if it isn't something that could be more easily deduced from the given information. I figure it must be something blindingly obvious if he didn't mention it...but I don't see it, my geometry knowledge being so poor.
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