Draw the tangents to the inscribed circle of triangle ABC that are parallel to the sides of ABC. These lines cut of 3 smaller triangles. The radii of the inscribed circles of the new triangles are 2, 3 and 10. Show that the triangle ABC is right-angled.
One thing I noticed that if we add up the radii in pairs we get 5, 12 and 13, which are the sides of a right angle triangle. Now look at the sides of the ABC triangle. If we could show that the segment from a vertex to the point where the inscribed circle of ABC touches the side is proportional to the smaller inscribed circle's radius, we would have an ABC triangle with sides 5x, 12x and 13x, which is a right angle triangle, and that is what we wanted to show. So what we need to show to complete the proof is that if we have a point and a circle, the tangent segments' length are proportional to the inscribed circle of the bigger circle and the two tangent lines.
EDIT: We also need to show that the inscribed circle touches the parallel lines at the same point as the smaller inscribed circles do, so that that the circles actually touch eachother.
One thing I noticed that if we add up the radii in pairs we get 5, 12 and 13, which are the sides of a right angle triangle. Now look at the sides of the ABC triangle. If we could show that the segment from a vertex to the point where the inscribed circle of ABC touches the side is proportional to the smaller inscribed circle's radius, we would have an ABC triangle with sides 5x, 12x and 13x, which is a right angle triangle, and that is what we wanted to show. So what we need to show to complete the proof is that if we have a point and a circle, the tangent segments' length are proportional to the inscribed circle of the bigger circle and the two tangent lines.
EDIT: We also need to show that the inscribed circle touches the parallel lines at the same point as the smaller inscribed circles do, so that that the circles actually touch eachother.
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