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?